The accomplishment of finishing games

Is finishing a game necessary to get the full enjoyment out of the game? Should a gamer even worry themselves about extracting all fun there is to be had from a game?

When I was much younger, I only got a few games a year. Some of these games, like Final Fantasy, were a lot of fun and also had a lot of time you could sink into them. Other games, like Jordan vs Bird, were far less fun to me. The thing is, while the games wouldn't always be amazing to play, my only other option, other than not playing any games at all, would be to revisit a game that I had likely beaten plenty of times by that point. So I stuck with the games, even the bad ones, and usually finished them despite their bugs, insane difficulties, and other flaws. And looking back, it was mostly fun doing so if only for the challenge and feeling of accomplishment.

So why now am I so likely to just abandon a game as soon as it fails to captivate me? I think quite simply, it is because I can reasonably say that I now have better things to do with my time and I'm capable of spending money in search of a new game that will succeed in captivating me once again. No longer being limited to two or three games per year enables me to continually seek out games that I will enjoy. However, I can't help but feel that by abandoning a game, I'm not playing games properly. Can I really say that I played Bioshock if I only played through two-thirds of the game? Can I claim that I enjoyed it, which I did, even though I became somewhat bored with it and moved on to the next shiny thing?

Maybe I worry too much about stupid things. Maybe I've forgotten how great that feeling of accomplishment can be.