Can you sneak attack and rage at the same time?
Honestly, I only ask because it seems like one of those things that you just shouldn't be able to do, but I can't find any rule preventing it. Am I missing something?
Yes, of course
Rage specifies that you cannot do anything that requires patience or concentration, but it does not eliminate your ability to attack accurately (it actually enhances that), nor does it eliminate your knowledge of taking foes unaware. The Sneak Attack ability represents the fact that a rogue has practiced taking advantage of vulnerabilities so much that it is a second-nature to him: as long as the opponent is distracted, he can take full advantage. He does not have to think about it or concentrate on landing the blow in the perfect spot: that is a thing he does naturally by now. So raging has no negative impact on his ability to do so.
You seem to be thinking of Sneak Attack as something more like the Assassin’s Death Attack: studying an opponent, waiting for an absolutely perfect opportunity. Most would probably rule that Rage prevents Death Attack. But Sneak Attack is not nearly as careful; it is taking merciless advantage of all but the most slight of opportunities. All you need is an opponent is not paying quite as much attention to you as he should be.
If balance is your concern, I wouldn’t worry about it. A Barbarian can do a lot more damage by focusing on his Strength for attack, damage, and more damage from Power Attack, and then using damage multipliers like lance-charging. Sneak Attack is best when you can apply it repeatedly, i.e. a full-attack while dual-wielding. Mixing the two means you’ll be less-good at both than someone dedicated. Mathematically, most of the time you’d be better off specializing (and most of the time, charging is the better specialization). The combination can work alright, but it’s nothing like overpowered at most tables. At many tables it will be underpowered.
Since concerns of imagery have been brought up, I think it’s instructive to think of the tiger. The world’s largest cat, the tiger is huge and brutal. It’s also terrifyingly stealthy, difficult to see and harder to hear. It sneaks up to prey and pounces at an opportune moment, and then proceeds to tear into the poor critter in a frenzy of claws and fangs. And yet, it still manages to go for the target’s throat, or to hamstring prey that attempts to run, very much “Sneak Attack”-esque maneuvers. Think of your Barbarian/Rogue as a metaphorical tiger, and it should not be hard to imagine how he Sneak Attacks during a Rage.
In addition to the prior answers, I think the most important circumstances for a sneak attack in this situation are as follows:
When raging, a barbarian wants to go for the throat implicitly, so when the unlucky recipient can't properly sidestep the attack it is all the more likely to strike home with the misguiding name of "Sneak Attack". Normally, "flat-footed" is tantamount to "unaware" in a lot of people's eyes which is where the problem occurs. "Flat footed" can also describe that 'deer in headlights' moment when the hulking barbarian is bearing down on them.
Yes, you can Sneak Attack while raging. As @KRyan has exhaustively explained, this need not be unreasonable.
The reason this is mechanically possible is that Rage explicitly lists the activities that are impossible to perform while raging.
As you can see, "the Sneak Attack class feature" is not among the actions prohibited. You would have to make a case for Sneak Attack requiring "patience or concentration" if you wanted to forbid it, which I would call a stretch. Refer to @KRyan's post.
Further, the Sneak Attack class feature also has a small list of circumstances under which it can not be used:
Again, Rage is not on the list. I conclude that it works fine.