A contraction of "programming"/"program" and "problems". No program ever created has ever been problem free, and thus has proglems of varying degrees of stupidity from bad coders.
Software Developer: We just inherited this program from an indian company. It's terrible and full of proglems, they recreated the same page 3 different times for 3 different log ons rather than reusing the same page and hiding things based on the users authenticated role.
Coworker: Sucks to be you, they're your proglems now.
Software Developer: :(