The speed at which you can competitively play first-person-shooter games depends on many factors, but the one most overlooked is the lad time of the display. FPS PC gamers are always on the hunt for an advantage, so when specialized gaming monitors were first released years ago with response speeds up to 1 or 2ms, they became the next big thing in the gaming world despite tradeoffs in reduced color accuracy and poor off axis viewing angles. While most LCD monitors have improved over the years and some have become very fast, not everyone has the latest and greatest of displays, especially when it comes to televisions which prioritize the best possible picture quality over the fastest response time. This will be more noticeable to PC gamers that have since come to consoles and use an HDTV for their display, since monitors are almost always faster than TV displays. You can however improve your TV gaming by making sure to use the TV's "Game Mode" if it has one, and turn off all picture processing features since they just add to lag issues.
If you happen to be in the market for a new monitor or TV, this site has some pretty good information.