The class numbers of the real cyclotomic fields ${\bf Q}(\cos(2\pi/{p^n}))$ are very difficult to compute. Indeed, they are not known for a single prime $p>67$. We analyze these class numbers using the Cohen-Lenstra heuristics on class groups and are led to make the following conjecture: For all but finitely many primes $p$, the class number of ${\bf Q}(\cos(2\pi/ {p^n}))$ is equal to the class number of ${\bf Q}(\cos(2\pi/ {p}))$ for all positive integers $n$. It is possible that there are no exceptional primes $p$ at all. Work in progress to test this conjecture empirically will also be discussed. This is joint work with Joe Buhler and Carl Pomerance.