Is there a limit to the number of Perl modules one can use in a Perl script?

That's limited, in practical terms, only by the amount of memory you want to use. Perl imposes no hard limit.

If one uses a lot of modules, does that slow a script down when it is run?

Not under most circumstances. The factorization and compartmentation modules encourage is likely to improve overall efficiency. There is some time overhead on sub calls, and using lots of memory can bring a script to its knees.