There are many formulations of the PSR, just as there are many formulations of just about any other principle or argument in philosophy. Some versions seem very implausible, while others are more well-accepted. The short answer is that no formulation is universally accepted - so yes, it is certainly possible that the PSR, in any of its formulations, is false, just as it is possible that any of the formulations is true. But I'll give a few examples of what I'm talking about.*
One version of the PSR, inspired by Spinoza and Leibniz, says something to this effect: "for everything that is so, there is a sufficient reason for its being so." That is, every proposition has an explanation that is sufficient (that is, it accounts for what is being explained, and it also rules out the possibility of some state of affairs other than that being explained). This version makes a very strong claim, and so has not been very widely accepted.
Another formulation, something that someone like St. Thomas Aquinas might defend, goes like this: "for everything that exists, there is an explanation for its existence." Notice that this version of the PSR makes a much weaker claim, namely that only the existence of every being needs explanation, and not that the truth of every true proposition needs an explanation. This version is, to me and to other philosophers, far more plausible, though it is by no means accepted universally.
I will also make a brief remark in disagreement with Yannik Ammann's answer. Under any formulation of the PSR that I have encountered, the fact that a being is necessary is an explanation of its existence. In fact, the entire point of the arguments for the existence of God of which you spoke in your question is to prove, from the premise that there are contingent beings and the premise that the existence of those beings needs an explanation, that there is a necessary being that serves as a kind of ultimate explanation, and which contains the explanation of its own existence in itself. So, whatever the other faults of the PSR may be, the possibility that it eventually runs into a "but what about God?" objection is not one of them.
*I first came across these two formulations of the PSR in Peter van Inwagen's Metaphysics.