This can be a very difficult problem in practice, depending on your expectations. You would firstly need to assume that in no games were there blunders, and already you can see problems begin to crop up as it is difficult to define what constitutes a blunder (Win to loss? Win to draw? Queen blunder? Positional blunder? Touch-move?).
If you make the further assumption that all of a player's opponents are 'fairly' rated (although this is something fairly dubious to say), then it may be somewhat possible to come up with statistical tests. But first off, it is better to rephrase the question from "X rating points in Y games" to "X game points from Y games", because rating changes depend on the K-value.
Consider the following: you have n games, and are expected to win "rating change"/"K-value" of them. You can also compute the expected score for each game (which depends on rating difference), and use these as your probabilities in a sequence of multinomial trials. For example, say you have 5 opponents, with rating difference relative to you and "win expected" as follows:
Then your total wins expected is 2.32. Constructing a statistical test here is tricky, because you can't use the binomial distribution properly (as it assumes that only two results are possible, whereas in practice, a third result, a draw, is also possible), and the Elo rating system only gives you the expected score, and not the win/draw/loss probabilities, which depends on the absolute value of the players' rating (i.e. a draw is much more likely between 2700 vs. 2800 than 1700 vs. 1800) as well as colours (whose significance, again, depends on absolute rating)!
You could, of course, go for a naive test and ignore the final factors, but if your total score contains a half point, you won't even be able to use this test!
Anyway, supposing you have the easy case of a sequence of wins/losses without draws, say you scored 3/5 in the above example. Then for your p-value, you would need to compute the probability of scoring 3 points or more, which you get by summing over the probabilities of all possible ways to score 3 points or more (wins vs. players 123, 124, ..., 345, 1234,..., 2345, 12345) which would take quite some time to be done by hand (you need to calculate and sum 5C3+5C4+5C5=16 products of 5 numbers), and if this sum is less than some critical value, say 0.05, then according to this very naive test, your actual performance is indeed better than your rating suggests.
Actually, the test would not be too bad if you knew the drawing probabilities: you can do the above again, though there will be much more work; the assistance of a computer is definitely necessary.
At the end of the day, the best test is time and exposure. Play in international tournaments, and if you maintain your rating change over time, then your playing strength has certainly changed.
(Technically, the tests I described are one-tailed, and you might need to consider two-tailed tests, but deciding on which to use and their impact is relatively trivial and unimportant compared with the other problems you need to address!)