The short e and the short i can be very close in pronunciation.
I say "pen" and "pin" almost exactly the same way. If I think about the pronunciation very carefully, I can pronounce them a with some distinction, but not in normal conversation.
I pronounce PIN and PEN quite distinctly. I'm a New Zealander. I guess with some accents they are similar, but it's probably best to suggest only the proper pronunciation, i.e. GYM is pronounced like JIM.
A fellow by the name of Rick Aschmann has created and compiled a set of maps showing which regions of the US pronounce things which way ( http://aschmann.net/AmEng/ ). This pdf at the University of Pennsylvania has a map for pin/pen: (