For leagues on THS, I usually carry 4, because I have a 4 ball bag and 1 of them is a plastic ball for spares. If I can't figure out a house shot with 3 balls, something's wrong. Most of the time on a THS, I can use any ball in my bag, it's just a matter of where I put my feet, hand position, speed and target. But I'm the type of guy that will punch up a new ball just to see what it's about. On tougher conditions, matching up with the right ball is much more important because that's where you really see the difference between the different bowling balls in your bag. On a THS, the key for me most of the time is to have a ball that blends out the over/under that's very common on the league shots in Tucson, especially this year.
But I agree with dogman to a point, if you're bringing that many balls to league, most of the time you're going to end up out thinking yourself, which is a primary reason for not scoring well. (I'm definitely speaking from experience here.)