Reboots happen when people loose interest in the franchise and they need a way to reinvent it to make it popular again.  Games like Halo and Metal Gear Solid aren't going to get reboots (anytime soon) because their still hugely popular in their current form.  While not a videogame, look at Star Trek.  10 movies, 5 TV shows and countless videogames and novels before they needed to reboot the thing.  Although in that case it wasn't so much the lack of interest in the series but that the current producers had utterly ruined it beyond salvation.

No, I don't think this is a solid theory at all. There are a lot of factors that come into play when deciding when a series needs a reboot such as time between games, time since the series launched, evolution of the genre since the series launched, how well received the series has been over time, and other factors. Also I don't think you could almost call Halo 3: ODST a reboot of Halo. It ran in the Halo 3 engine with almost all of the weapons, vehicles, and enemies from the previous games, and tied itself into a major story event from the Halo trilogy.

" I think you're clutching at straws with this theory... "

I read your theory then look at Metal Gear Solid and just laugh.

OH GOD DAMMIT!!!

With the exception of DmC, none of what you listed I would call a reboot by any stretch of the definition. A reboot is meant to take a narrative back to it's original start with a new twist. It's meant to ignore everything that has already been established and start fresh. 
 
Also a reboot wouldn't have a number in it, least of all a higher number than it's predecessor.  Also you have to count hand held sequels as well.
 
Civ 5 might be streamlined, but that's not what makes a reboot.  
ODST certainly was not a reboot at all. Sure it had a different mechanic and it focused on a separate team. But it's a interlude to the main story. 
Why are you discounting the first COD? That was made by infinity ward just like 2, 4, and MW2. It should count. And there was no single narrative in COD until COD4. But COD4 was not meant to be a reboot, just a continuation. Hence why it's called the fourth game in the COD franchise.
Ratchet and Clank went 5 games before moving into it's "Future Trilogy." Size Matters is titled Ratchet and Clank 5 in some countries, so it counts. Also I can't see how the Future games are at all a reboot. All the Ratchet and Clank games are pretty identical in style and game play.
If Splinter Cell Conviction was meant to be a reboot, then they wouldn't have it take place after the last game.
 
If you want to know what a reboot actually is, look at Casino Royale or Batman Begins. Both movies have the character start from their absolute beginnings. Ignoring everything that happened before so that the can breathe new life into an old character. None of your examples fit what a reboot is. 

The term reboot, in media dealing with serial fiction, means to discard much or even all previous continuity in the series and start a new with fresh ideas.Effectively, all established fictive history is declared by the writer(s) to be null and void, or at least irrelevant to the new storyline, and the series starts over as if brand-new

" ...this is the fourth Tin-Foil Hat thread this night.  O_o  It's like a conspiracy night or something. "

NuMb3rZ   AHHHHHHHJHH

...this is the fourth Tin-Foil Hat thread this night.  O_o 
 
It's like a conspiracy night or something.

it takes around 2 to 4 years to make a good sequel. system Generations only last about 10. Most mid-teir games (games that always do well, but never mega sell like the Halo's and such) use any tactic they can to get people to buy their game.  
 as a result, we get sub titles and whatever else every 4 games or so.  Its not so much that 3 is the magic number.  
 But there's more to it. Most contracts for these sorts of things only cover x-amount of years or titles. Just like actors getting signed for 3 movies, developers tend to pick people up for 3 or 4 games, or 3  or 4 years. Bungie made Reach different to show that they werent limited to the Halo build of lone super-gunman heroics.  
 
there are allot of factors, but the number 3 isnt just flat out the cause. Thats Hollywood thinking...

A conspiracy!

 Was just listening to the Bonus EXP podcast, and they were talking about Civ 5 a bit. Andy basically said he already had 4, so why bother? 
It got me thinking about franchise reboots or retools, and how many games a series can go before it's time for a reboot. Based on the few games I was thinking about, it seems like 4 is a good number. 
A few games I was thinking of: 
Devil May Cry went to 4, and is now coming back as just Devil May Cry again. 
Civ 4 to Civ 5 
Whatever the next Halo game is (you could almost count odst as a reboot, except they really didn't go anywhere with it, and essentially finished a story arc with Reach) 
Call of Duty 3 to 4 (I'm discounting the 1st CoD, and the portable releases in this case) 
Ratchet and Clank: went four games, and rebooted as Ratchet and Clank future, which is on tap to get a 4th game in the series. 
Splinter Cell 
 
Obviously there's franchises I'm not familiar with that may add or subtract to this idea, but I'm curious to know your thoughts on if if there's a reason 4 seems to pop up the way it does, and if there are games that maybe you wish would have had a reboot that didn't get one.