Well, after the success of
The Bourne Identity, some guy picked up the series and wrote a fourth installment. This one took place like 20 years after the previous one had, so presumably you could do that and use it as an excuse to recast Damon's role.
However, I've also heard that the two sequels to be written since
Ultimatum are weak at best.
Then again, the film versions wildly differ from the novels, so maybe a fourth installment could work...
...but then again, three is a fine number to end the series on...
...personally, I think they should just leave it be. Three decent movies that tell a pretty cool story. I think that's a good to leave it before you get too overzealous.
The Vile One said:
4th installments are rare no longer what with Live Free or Die Hard. Indy 4 next year. And even Beverly Hills Cop 4 around the corner.
4th are in now.
These are also sequels to film franchises that have been unused for a couple of decades. It's not so much that "The Fourth One" is in now, I think, so much as it is that Hollywood is trying to revive all of these old franchises which just happened to end on the third installment.
Plus, as of yet the only "#4" that's been released that I can think of is
Live Free or Die Hard, so that's our only measuring stick. It might be wise to wait until more sequels come out before the legitimacy of the practice is decided.