This is a dangerous misinterpretation of p values, which cannot provide that kind of information. A p value assumes the pattern is due to random fluctuations, and asks how common this kind of fluctuation is.
Typically the chance the result is a random fluctuation is much higher; for examples, see http://www.statisticsdonewrong.com/p-value.html
If you have a test of significance that results in p < 0.01, there's a one percent chance that you're rejecting the null hypothesis due to normally-distributed variation in your data. The base rate fallacy is more about interpreting what that p = 0.01 means, and why systematic bias is important to worry about - if you're testing cancer drugs, you don't want to test them on people who don't have cancer.
No, this is absolutely not true. If p < 0.01, then if there is no systematic effect and only normally-distributed variation, you would see this effect 1% of the time. That is, the p is P(data | null is true), and not P(null is true | data). You cannot invert the conditional.
In the extreme case, when the null is true for every test, you will get significant results for 5% of them. Thus 100% of your statistically significant results are false positives, no matter how small their p values.
Given that we do not know what fraction of the time the null is true, we cannot know the chance that we're rejecting the null falsely. But it is invariably larger than p.
This misunderstanding is why scientists routinely overestimate the strength of their evidence and discount the possibility that their results may be flukes.
(Source: I wrote the link provided earlier. Also, the discussion leading to table 1 in this paper is good http://journals.plos.org/plosmedicine/article?id=10.1371/jou...)
The problem is that the statistical model (in my field we do a lot of ANOVA and t-tests, along with the occasional chi-square) can only account for what you model. So there could be some kind of systematic error that influences your results in a fashion that is not modeled by the statistics. Having a large-N study makes it harder to have that systematic error (but not impossible - as an example: look at complaints about how much psychological and cognitive science research is only on WEIRD subjects - western, educated, industrial, rich, developed).
The other problem, of course, is that one time in a hundred, you'll get a p < 0.01 significant result by chance. Which is a lot in the long run. Worse, you can induce type two errors by running hundreds of trials (or testing hundreds of variables) and not accounting for that - just pick the one thing that had significant results on a single test. This approach is unscrupulous, but not unheard of in academic circles where you need to publish tons of work to get promoted.