The NSAID's (Non-Steroidal Anti-Inflammatory Drugs) - as the name implies - decrease the inflammation response to injuries, and strength training deals with damaging tissues and then repairing them. So if you are on 6 Aleve's every day, that probably will have a negative impact on your gains. IMO, I treat Ibuprofen, Aleve, and Aspirin (even though it's a little different) the same. They tend to be relatively harmless for the liver in moderate doses. They have been shown to be hard on your stomach and kidneys.
Tylenol works differenly and doesn't work as much on the inflammation, but works on the nerve's ability to signal pain and it also does a better job at fever reduction. It's much harder on the liver and even with all the pull the companies have that produce it, things are bad enough they are reducing the recommended amounts and so-forth, especially in Europe were deaths are high from Tylenol overdose. Admittedly, Oxycodone / Oxycontin aren't great if you get hooked on them, but they tend to not be toxic over the longer-run. So as long as it clears your system and you don't OD, the cumulative damage isn't bad and it doesn't affect inflammation - which is natural for the body to recover.
What's even worse than NSAID's are steroid injections (not AAS, but anti-inflammatory - like cortisone if taken constantly. An acute injury can be helped by using either NSAID's or the steroids, but mainly with pain and only right after injury. Otherwise, if you continue to take NSAID's, tissues just don't repair as fast or maybe not as well once complete....they even have an effect on the healing of broken bones.
Anyway, if you have a headache once a month, taking something for it isn't going to affect gains. Taking 4 Aleve's everyday will probably hurt gains - much more so if you are natural. Also, one of the first signals that you are taking too much NSAID's is heartburn that you didn't have before. Tylenol doesn't affect the inflammation as much - which, in contrast to what you've always heard, is good. Inflammation signals for the body to repair.
Overall, I don't think the benefits of taking NSAIDs often match the costs. If you are taking them often, probably need to do something to correct the issue. And although the opiates don't affect the inflammation significantly, if you are natural, you should know that opiates (if used consistently) lower testosterone production. The absolute worst time to take opiate pain killers would be during PCT.
Lot of rambling there...maybe something you were wanting to know???
Good luck.