Please explain if it is possible for a pain medication to have an 'anti' affect, make the pain worse?

Pain meds. Anti analgesic effect after long term intake of high dose opiates is a well known phenomenon. Start tapering under medical direction.
Of course. it's called medication overuse pain. The good example is medication overuse headache. It is more likely to happen with butalbital, NSAIDs and opioids. It is often associated with headache, muscle pain, nonrestorative sleep, neck pain, and vasomotor instability. See your treating physician for medication adjustment.