IIRC, there was such a result as "there is no more than 1 non-trivial solution of $x^n+y^n=z^n$, if any", wasn't it? (IIRC, Siegel theorem implies that there are finitely many solutions for $n>3$; so it is the "no more than 1" part that is of particular interest).
Also, any reviews of pre-Wiles' results on Fermat's Last Theorem are appreciated.
The Wikipedia article gives a good summary of pre-Wiles work. One highlight: In 1985, Leonard Adleman, Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes $p$.