I know that FLT took so long because we hadn't had tools to solve the problem at the time it became known.
Upon reading Wikipedia page on Basel problem it appear that same was the reason for it too by the sentence-
His[Euler's] arguments were based on manipulations that were not justified at the time, although he was later proven correct, and it was not until 1741 that he was able to produce a truly rigorous proof.
So, is it always the case that if a problem take so long to solve, it is only due to the reason that we developed the necessary tools meanwhile?
One can also answer the question by providing example(s) of mathematical problems which stayed unsolved for decades after formulation despite the fact that all the necessary methods had already existed at the time. In this way it can be understood that there are other reasons too like our ignorance.
The question of whether there is a polynomial-time algorithm for testing primality was around for a few decades before it was settled by the AKS-algorithm, which used no mathematics that wasn't already known at the time the question was formulated. Here's a reference.