Sometimes (as a player) I get bored during combat. At times I feel like all my turns are the same (especially when playing lower level characters) and when that is combined with no combat progress (adventurers are just whiffing round after round) things can get dull. What can as a player do to "shake things up" and fight my boredom?
Do something other than hit-roll-for-damage. Think of it as a movie. First and foremost, just do what you want to do. Worry about the rules second.
As a GM, my NPCs will use fire, explosives or other special weapons to divide the players or remove their protective location. Fleeing may cause the PCs to chase haphazardly, triggering traps. Sometimes they just do something stupid which causes untold chaos (such as using a sonic weapon in a cave, causing a cave-in.) There's no reason PCs couldn't do the same.
The most fun combat characters I've seen are acrobatic melee fighters. I often provided multi-level arenas, and one of the PCs made good use of it. He frequently launched himself off the back of moving vehicles, clung to the underside of platforms, and threw people to their demise. He took down a helicopter in melee for f's sake! There's no rules for half the stuff he does. I make a call, give him a +/- as appropriate, and if he concurs he goes ahead with the manuveur.
Another PC who had a lot of fun was the pacifist. He never broke character. He used stealth a lot and only stuned opponents. He even offered healing once they were subdued. When it came to fighting mechanised opponents, he located weakpoints for the benefit of the team and jury-rigged fuel bombs. It forced him to think creatively about his tactics.
And then one PC was not-quite-Wolverine, built entirely for combat. Sure, he basically played hit-roll-for-damage, but he narated every move and loved every gorey moment of it.
Oh, and we draw all of the blood on the map, especially if there's a chance we'll return to this location.
Your issue with the "whiffing" part of low-level D&D-ish play¹ is solvable by a different choice of game, or by starting at levels where fighters are competent. More on this below.
Note that D&D isn't really realistic if you describe the failed to-hit as a "miss" - D&D is, at its core, conflating hitting and penetrating the armor into a single roll. It's important to realize that D&D "to-hit" is actually "to penetrate the armor." And that a high damage roll can be one big telling hit, or several smaller penetrating hits. Getting the GM to narrate this type of situation changes the feel considerably for some players. It's also more work.
When playing, you as a player can avoid the mechanical crunch by doing off-the-wall acts of derring-do... rather than simply attacking, try dropping chandeliers on people. Or fighting from cover. Find ways to get the character action into narrative mode rather than simple "minis battle" mode.
And, of course, if the combat looks like it will be too long, encourage the others to join you in evading that encounter, or doing "stupid-player-tricks" to get the bad guys to go away rather than fighting it out. Heck, sometimes surrender is more fun than a long battle. See if there is overhead stuff. Use useful illusions... like having the lead fighter looking like he's under the influence of brain parasites - you know, the ones that live by boring through the skull and leaving spicules outside the skull - or has nasty virulent leprosy, including suppurating wounds.
In other words, Play smarter. Either your GM will to, or he'll attempt to nerf you. Either way, you get a strong read on the GM, and you let the GM know you don't like long fights.
No. 5 may seem a bit radical, but it's important to tell the GM when he's boring you. Just pass him a note - don't publicly call him on it. And for heaven's sake, don't just dive into some other activity between turns.
Below (Other games)
Now, I said more on other games below - here's below...
You're obviously used to Class and Level based games. Within that framework, there are several that lack the "whiff" factor D&D builds in.
Palladium, much derided for several issues, actually has a decent combat model. The to-hit roll is actually pretty low - 5+ on 1d20, and bonuses to hit from skill apply (tho' a nat 1 always misses). The defender gets to dodge or parry, with a TN of your to hit roll. And the low-tech armor rules mean that the armor absorbs much of the damage - if the attacker's roll was 5+ but still less than the AR, it still hits, but does the damage to the armor. It's no faster than D&D, but you get to roll when it's not your turn, and you don't take all your actions at once.
In general, tho', Class and Level leads to Whiff in many cases. The others are, like Palladium, skill based, but generally, Palladium is the least flexible. Also the fastest playing, in my experience.
Rolemaster has more HP at start than D&D - and they're needed, as lots of minor damage is done. But, the HP go up slightly slower, and the combat is resolved more quickly as damage leads to critical damage, and good crits can kill. Plus the crit tables are fun. Very little whiff. Table heavy, math intensive game, tho'.
Original (little-book) D&D with the minis combat system is actually shockingly fast. The d20-driven "Alternate Combat System" actually reduces the effectiveness of combattants drastically - so return to the original, and see a vast improvement in combat speed... and thus reduction of down time.
Before you complain...
Be certain that you aren't someone else's problem, too.
If combat drags, make certain you're ready when your turn comes. You, as a player, can encourage others to up the pace, too. It can be infectious.
¹: D&D, a retroclone, or d20 or a d20 derived game... which, from the terms used, and the whiffing issue, are pretty clear giveaways of D&D or a close relative. Many other Class & Level systems don't have the whiff factor.
Depending on which game you are playing, combat literally can just be doing the same thing because that's where the bonuses are. Other times, it's because players aren't ambitious with the rules that are present. Taking the AEG games into example (L5R/7th Sea) there is a lot of opening to mix things up by making your attacks more difficult with "Raises". A good 3/4 of the time, my players don't use them even in situations that I think I've made obvious that they should. For example, practically stomping my foot when I say "now's your chance to do damage" and the brute just makes a regular, mundane attack.
Then there's the third category where no matter how exciting the system is, there just simply isn't anything that fits your character's niche and you wind up throwing your dice, probably not even totaling the result beyond a visual check of high/low. To which I might suggest the GM have a couple of NPCs around to let the occluded player use for the fight (whether working for or against the party). Because of a certain level of meta-gaming, all characters tend to be in a vast majority of encounters, which is why something like this happens. Why is the social character in a dungeon? Ex Machina. Why is the brute without manners in the highfalutin' party? Ex Machina.
As a catch-all solution for any given game, I recommend talking to your GM about instituting a time limit, say five seconds or based on their Intelligence/Wits (faster thinking in game, a little more time to think out of game), to start declaring what a given character is doing. Some players (occasionally guilty) like to plot out four moves in advance, count, recount, and that's not combat. They are already being given entire minutes to consider their actions in what is already a Charlie Foxtrot of a situation so if they don't have the answer, they get skipped either by holding the action or whatever the game determines is the default for doing nothing.
This is a question I wished some of my fellow players would ask themselves. They always start playing with their phones when its not their turn in combat, and then extend combat because they have to see what happened and then start making a choice about what to do after all that.
Personally I try to work out what might happen as things progress and try to have different tactical options in my head. I also think about each other character's abilities as their turns come up to try to remind them of feats they may forget. We play on a whiteboard so doodling the battle itself out can be a pastime as well. Lastly, try taking notes but with the approach of a less than truthful/over-enthusiastic chronicler and write it for comedy. It'll give you something to do, but still focus you on whats happening.
Fight boredom in combat by making it more narrative. "I get stabby with +3 sword" invites boredom, however "I make a bold thrust using my favorite sword, Vera" sounds much more interesting, but mechanically you are using the same sword, and making a "general" attack. I once had to do a 1-on-1 combat where we had to pantomime the whole thing. It was very memorable. Look up fencing and other combat terms to help your descriptions.
One rule I've heard about (but haven't tried) is that you have each round go twice. The first part is a very quick and mechanical (rolling normal attack, hit AC2 for 10 damage), then the effects all take place narratively. Either the DM does it all them self, or the players each say their actions in very narrative text (for example, "I took a bold feint to the enemy's head, only to dip my sword at the last minute, catching him in the mid section for a significant wound"). It will slow down combat, but should make it more interesting.
In oWod another trick we used was to take the extra successes in the to-hit roll and apply them as damage. I suppose you could also do the same thing in D&D, but I haven't tried it. This bonus damage can help make combat more interesting since your damage calculation changes every swing (although not recommended if your group drinks enough to make quick math calculations difficult).
Finally, use the tools you have. Let the brute hit in the first round (or 2) of combat, while you are setting up flanking maneuvers. Read the combat section of the rulebook to learn all the unusual maneuvers available to you, write them down and go through each of them. However, if you are going to do this, make sure your list also contains the page number and rulebook (or a very condensed version of the mechanics) so you know what needs to be rolled and the modifiers are at hand for the DM to apply as s/he sees fit. Also study various tactics that the masters have described in their books (Machivelli, Sun Tzu, et al.).
One method I use to avoid being bored by overly long combat is to end it quickly. This involves the following two steps. Warning: You have to do both of them, else it might backfire horribly.