He's dropped the gloves 10 times this year, the leader has 15 fights, and he's only gone two games this year without a hit. (which is still bad for his role)
Mind you, still far from 'many' like you claim.
Aside from a lower body injury suffered against Washington when he missed a few games, I recall twice seeing Neil getting visibly hurt although he continued to play. Once he took a hard hit mid chest but didn't go down. He labored the rest of the game and it was obvious he stopped trying to hit. Another time he skated hard on the forecheck and fell badly into the boards. It looked like he had injured his shoulder. He went off the ice for a while, came back but he wasn't effective. Both times for the next few games he didn't fight or even hit much. I was sure he was playing injured. I can't recall the game but in a post game interview, one guy said he asked Neil to fight and Neil told him he couldn't because he was injured.
I'm not saying he hasn't slowed down but sometimes there are reasons guys don't play with the same determination as they usually might. That said I don't think Neil is effective in his role unless he's 100% and the tough guy mentality or fear of losing one's spot may play into guys playing hurt. The NHL and teams should encourage guys to sit out when they have bobo's and make it cap easy to bring up replacement prospects as needed.