Let me play devil's advocate here (since everyone seems to take the negative position) and say yes, you absolutely can.
But, and this is a big but (no pun intended) the games you play have to be "good" games, and it will not be enough to simply play them, but rigorous analysis must also be done on each game that is played.
The absolute truth of the matter is that any student, with enough motivation, a cheap computer with a decent processor, some (free) chess software and either a lot of talent or a lot of time, can become an IM or even GM. This is something not many players will admit, because it seems to subvert all the chess-related purchases they might have made.
In the past, it was necessary to hire a tutor, or read books and other chess literature, to keep up-to-date on opening theory and to learn chess. Today, with the advent of powerful chess engines and game databases, much of this is obsolete. Furthermore, I would argue that chess books are not only outdated, but outright inefficient when compared to the alternatives. These are bold claims, so let me try and support them with a plan of action which is bereft of any chess literature and focuses only on games.
- Don't play 100 short games a day, this is a waste of time. Play 1 long game (90 minutes or more) against a moderately strong computer engine and spend the rest of the time doing analysis
- You will lose pretty much every game you play (if the computer is set to its highest level) but you must not get discouraged or change your style of play to beat the computer (i.e don't play closed positions solely for the reason that computers are bad with them). This is where motivation and discipline will determine whether you really benefit from this method.
- In every game, focus on a single thing and try to improve it. Initially the biggest problem might be gross blunders, hence you should do a blunder check on every move you play to prevent such blunders from occurring.
- You should try to create a thinking algorithm, a step-by-step process or checklist which helps you to make decisions over-the-board and then use this algorithm for every move (until it becomes habitual and automatic). Steps might include things like the blunder-check above, or deciding whether the position is tactical (requires intensive calculation) or positional (requires sound planning and positional understanding). This is the only thing where it might be faster (not necessarily better) to read up on, instead of building it yourself, since efficient thinking systems already exist that were created by very successful players (Capablanca, Nimzovitch, Tal etc.).
- Analysis of your games should be done to figure out weaknesses or lapses in understanding. The easiest way to do this is to look at positions where you spent the most amount of time. A computer must be used to keep yourself objective and grounded in the concrete, but is used only to check your analysis, not to replace it. (Most computer engines have the option to analyze your annotations and almost all give a point score for every move, take advantage of these options).
- Play in tournaments and OTB games, record them, and analyze them the same as your other games. These games are more valuable, because you can actually win them once in a while and because they are games outside of a controlled environment (i.e not in the comfort of your home).
- You will undoubtedly improve the most by focusing on your own games, but for a change of pace you might want to analyze a collection of games by a strong GM or favorite player. You should do this by playing guess-the-move from the winning side, making annotations for moves you guess (correctly or incorrectly). Then check the game with a computer engine (you'd be surprised with how many mistakes you find, both in your own analysis and in the moves of the GM's). You can also use these games to figure out the thinking algorithms of the masters.
If you actually stick to the above recommendations, I really think you'd make significant progress. Much more so than if you spend an equivalent amount of time reading a book. The reason is because passively reading something is very inefficient, compared to actively learning through difficult training games. Why is reading/chess literature still popular? Mostly because it gives a false sense of accomplishment and because its enjoyable. Ask yourself if the last chess book you read gave you a significant boost to your rating or chess understanding. Ask yourself how much you actually remember from the book, whether the concepts or games or annotations, and how much of it has actually influenced your OTB play.
The method above is not merely my own conjecture but is inspired by the training system developed by Botvinnik (there is a portion I left out, concerning physical exercise, but I think that's self-evident), who emphasized long-time-control training games and thorough analysis for chess improvement. The emphasis on thinking algorithims comes from the research done by de Groot.
Ultimately the final verdict, is yes, you can improve in chess through just games. But few have the dedication, discipline and motivation to do so.