To quote perlvar:

You shouldn't depend on the value of $! to be anything in particular unless you've gotten a specific error return indicating a system error.

`` is not a system call (it's an operator), and the documentation of the operator doesn't mention $!. You haven't been told there was an error (much less a system error) before checking $!. Therefore, it's a bug to check $!.

Other functions (such as open(FILE, '...|')) provide more feedback on errors.