Statements that seem true for many values of n but they are false

Question

Not much time ago I saw an Instagram post about something called Borwein Integral, which is certainly such a "monster". This integral has the following form: $$\int_0^{\infty}\frac{\sin{x}}{x}\cdots\frac{\sin{\frac{x}{100n+1}}}{\frac{x}{100n+1}}$$
The most surprising thing about this result is that is true for all $n$ smaller than a number which has more than 30 digits surely, what shows that in this beautiful science proving a proposition for each value of $n$ is not enough. Please, correct me if I am wrong. Anyway, my question is not related to this integral, but to other results that are true for many natural values (at least of the order of trillions) but false for bigger numbers. Are there any similar results? Taking into account the infinitude of maths, there must be other thousands of results like this one, and I was wondering if someone could tell me about other one like this.