Mystic Theurge is a Trap
(assuming you enter as the expected Wizard 3/Archivist 3)
Gaining 13th-level Archivist spellcasting is not worth three lost Wizard levels, and gaining 13th-level Wizard spellcasting is not worth three lost Archivist levels (actually, the situation at 20 is better than at most levels because you have 17th level on one side and therefore finally have the same level spells as a single-classed caster).
At any level, a single-classed spellcaster is better than any Mystic Theurge
(assuming a 3/3 entry)
Basically, a Wizard 16 or a Cleric 16 or an Archivist 16 is much better than a Wizard 3/Archivist 3/Mystic Theurge 10, and this is despite the fact that ECL 16 is the best level (other than perhaps ECL 20) for such a character. This is due basically entirely because spells grow better-than-exponentially in power. That makes having the highest-level spell you can at a given ECL basically the best route to power in the game.
Anyway, at all levels that aren’t ECL 16, you are proportionally further behind, being a minimum of 1 and sometimes 2 spell levels behind until ECL 20. At ECL 20 you have 17th-level spellcasting, i.e. 9th-level spells, on one side, same as a single-classed character, but 13th-level spellcasting, i.e. 7th-level spells, on the other side isn’t all that impressive. If you are starting at ECL 20 (or greater, I suppose), it might be worth considering.
Just in case you were thinking that Mystic Theurges did have some advantages, I’ll go through some of the common misconceptions about the class.
Fallacy: Mystic Theurges have a lot of spell slots
Mystic Theurges have a lot of lower level spell slots, but no higher level spell slots. The difference in total spell slots at each level is small, but the difference in spell level is large.
Fallacy: Mystic Theurges are especially versatile
There are spells that divine casters get that Wizards don’t and want (divine power, heal, miracle), and spells that arcanists get that Clerics don’t and want (contingency, time stop, shapechange, etc.; there are more on the Arcane side), but those are only a small subset; for the most part the lists have a lot of overlap, particularly on the best spells. There are other ways of getting the one or two great spells you’re missing (e.g. Time Domain gets Clerics both contingency and time stop, Arcane Disciple can get Wizards a select few divine spells, etc.). These spells can be very valuable, but the cost to get them through Mystic Theurge (i.e. 3 lost spellcasting levels) is immense.
Moreover, no matter what lower-level spells you’re getting by being a Mystic Theurge, the higher-level spells you would have had as a single-classed spellcaster would improve your versatility more. Those spells are more powerful and more flexible and open up new options for you, far more than the lower-level spells you get from the other class.
Multiple Ability Dependency: Not for Archivist/Wizard but other combos
Mystic Theurges usually cause issues where multiple ability scores become important, but Archivists are Int/Wis and Wizards are Int, so that’s OK. But for a Cleric/Wizard entry, this is a fairly serious problem, and eliminates a lot of the (small) edge that Mystic Theurges have in spell slots (since they have fewer bonus slots).
Level 17: What Now?
Your investment in another class comes up short at level 17, when you have no more Mystic Theurge levels to take and can no longer continue to advance both classes. This is what ultimately kills the utility of Mystic Theurge, because you never get even the theoretical return on investment that you might have gotten.
This does, admittedly, get better again at 20th when you “catch up” (to an extent) with your spellcasting on one side. But the vastly-lower spellcasting on the other side isn’t really doing that much for you at that point, which sort of begs the question: why’d you bother slogging your way to 20th in the first place? If you start at 20th level, this calculus does change a bit, but ultimately there are better ways to spend your levels.
If you can get in without 3 lost levels, it becomes a lot better
Two levels is probably still too many, but only one lost spellcasting level is looking pretty nice. There are ways to do this; Precocious Apprentice (Complete Arcane) is one common route. In this case, you only wind up with 11th-level spellcasting, i.e. 6th-level spells, which is not great, but you do lose only one spellcasting level, which is a lot better. Half the time (even levels), you have the same level spells as a single-classed character, and the other half the time you’re on par with a Sorcerer or Favored Soul anyway. Much better as an option if you don’t expect to ever have to deal with the “now what?” question caused by running out of Mystic Theurge levels.
Again, though, it does bear mentioning that there are more powerful things you could do with your levels. Not many, and most of those are probably inappropriate for most gaming tables (I’ve never actually played in a game where an Incantatrix or Initiate of the Sevenfold Veil would have been appropriate), but worth pointing out.
If you can do more than the typical spellcaster in 10 levels, that’s also good
Apostle of Peace (Book of Exalted Deeds), and Blighter and Ur-priest (both Complete Divine) gain 9th-level Divine spells in as many levels. Apostle of Peace and Blight have serious, game-disrupting requirements (complete party-wide non-violence for the Apostle, and frequent forest-razing for the Blighter) and are not recommended for any game, despite potential power. Ur-Priests cast off the Cleric list, and are relatively easy to make (note the Adaptation section that can even rid you of the Evil requirement), so they are usually the best bet.
Anyway, a Wizard who manages to meet the Fortitude requirements of Ur-priest can take two levels of it, and then Mystic Theurge between Ur-priest and Wizard for 8 levels. At the end of this, he has 10th-level Ur-priest spellcasting, i.e. 9th-level spells off the Cleric list, and his character level −2 for Wizard spellcasting. That’re pretty good. It’s not the most powerful thing you can do in the game, but it’re definitely up there, and almost certainly better than a single-classed Wizard at most levels.
On the flip side, a Cleric or Archivist can finagle their way into Sublime Chord to accomplish the same thing, though it’s harder since Sublime Chord is a better-designed class that makes these sorts of tricks difficult.
Note that fast-progression prestige classes, particularly in combination with dual-progression prestige classes, are liable to get you whacked with a Dungeon Master’s Guide. Any time you get a spell level before a Cleric or Wizard would, you’re causing trouble.
Wizards themselves figured this out eventually
Rare for Wizards to learn from past mistakes, but learn they did: later “theurge” prestige classes, i.e. those that progress two different spellcasting classes, can be entered with only one level on one side, or have class features, or both. They also stopped printing faster-progression prestige classes, which is also a wise move.
Anima Mage and Noctumancer (arcane+binding and spells+mysteries, respectively, Tome of Magic)
Arcane Hierophant (druid+arcane, Races of the Wild)
Jade Phoenix Mage and Ruby Knight Vindicator (arcane/divine+initiating, respectively, Tome of Battle)
Sapphire Hierarch and Soulcaster (divine/arcane+meldshaping, respectively, Magic of Incarnum)
Ultimate Magus (prepared+spontaneous arcane, Complete Mage)
These are prestige classes of this sort that “do it right.” In reality, Mystic Theurge is one of the only classes of this sort without any kind of class features.
Traps Can Be OK If You Know About Them
You lose a lot of power going Mystic Theurge. But you know what? Archivists and Clerics and Wizards have a lot of power to lose. These three (along with Artificer and Druid, who are more complicated to discuss here) are the “Tier-1” classes, the classes that can be all things to all people if you try hard enough. So even though a Mystic Theurge is a lot weaker than an Archivist or Cleric or Wizard, that’s not necessarily a bad thing. You’re still probably more powerful than any Core class besides Sorcerer. You just have to be aware of what you’re getting into.