Yes, jelly toys eventually break down. How you care for them can increase or decrease the speed at which this occurs. Storing them improperly is usually the cause, but exposure to heat, sunlight or fabrics can also cause them to degrade quickly.
If you are prone to infections or share your toys, condoms are an absolute must when considering a toy that is porous---not just jelly, but anything that is not plastic or silicone.
Storing porous toys in their original packaging is the best way to prevent this and you should always clean them thoroughly before and after use with a toy cleaner. I've found, in the past, that antibacterial soaps are actually too harsh for jelly toys.