A couple of thoughts from an FPS gamer that still plays FPS games and still does fine:
1. In the old days, most folks internet connections ranged from crap to fair. Currently, most are good. Pings are a real factor in FPS games, so if you had an advantage in the past, you may not be enjoying that advantage now.
2. We are at a point in the market where CPU's are not changing that rapidly. In the past if you had a Pentium II and everyone else had a lower level system, you would be at an advantage. Nowadays, most systems have pretty good graphics capabilities by default and you may not have a system advantage.
3. FPS games are nothing new. Everyone knows how to play them, and they haven't changed too much in the last 10 years. Everyone is using WASD instead of arrow keys, everyone knows about key-bindings, and how to find educational videos on YouTube that make you better. So skill levels have risen.
4. Resolutions are higher and mice are more accurate. As others have said, it was easier to hit the right pixel on 640x480 than on 1920 x 1280. There were a lot less of them. Unless you play Huntsman in TF2...
5. The games themselves aren't changing as dramatically. Most games have settled into the same scenario and controls, so your skills are transferable: Battlefield 4, COD 4, TF2, Arma III, CS:GO etc. Those game franchises have been out for ten or so years, so you have all kinds of folks who have played them a LONG time. No place for dominant casuals anymore...
6. Young people aren't better, they just have the time it requires to maintain skills at a high level. Now, this is contentious, but I played the hell out of some games and when I was at my peak, I was untouchable. This was in my 30's, and usually drunk. Now, I would walk into the same game and get owned AND REALLY PISSED OFF if I was 12 - 25 or so. I wouldn't say "damn, I'm old." I'd say, wish I had time to get back into form.
You can be better, it just takes more effort than it used to because skills are higher. Either way, don't believe that bullshit that your skills diminish in your 30's. This isn't running a triathlon, for God's sake.