We show that the p-adic valuation (where p is a given prime number) of certain rational numbers is unusually large. This generalizes very recent results of the author and of Dubickas, which are both related to the special case p = 2. The crucial point for obtaining our main result is the fact that the p-adic valuation of the rational numbers in question is unbounded from above. We confirm this fact by three different methods; the first two are elementary, while the third one relies on p-adic analysis.