It becomes too much when you realize that the next you ball you buy does not fill a gap in your arsenal. But getting to this point can take a long time, and actually knowing the equipment you have in store. I sometimes have the impression that "ball whores" to not care about this aspect at all, just punching up the next ball that "hooks more" or might yield that accidental 300 game which makes this orb the best thing since sliced bread.
Personally, I'll admit that I used to buy a lot of balls - too many than actually needed - but it had the poistive side effect of getting to know different manufacturers and ball ball concepts. Even though I consider owning more than 10 balls for a hobby bowler like me ridiculous, that "phase" made me wiser and more knowledgeable as a player.
Today I'd say that a full arsenal can be made from 5 or six balls, and then you can cover anything, with some overlap and potentially different reaction shapes. Just getting there is a tedious task, with some trail and error, and pointlessly punched holes.
Besides, I find less and less new equipment to try, which is also a budget problem - a high end ball, ready to roll, costs about US$350 here. I am rather looking for the ball that actually fits to my needs than falling for the "oh, goes even longer AND harder than last year's killer ball" delusion (see above). The romance is gone...