I do not have a definite answer, but in view of the discusion, and since it connects, as requested, the question to problems investigated in the literature, some remarks.

I claim that:

If an answer to this question is *known*, then it is of the form that there are infinitely many solutions.

This might seem like a strange claim, and I do *not* claim that the number of solutions is infinite, but here is the reason:

Erdős and Moser (in the 50s) conjectured that the equation
$$ 1^k + \dots + m^k = (m+1)^k $$
has only the solution $1+2=3$.

Over the decades it was investigated quite a bit, still it is *unknown* whether the number of solutions to this equation (the special case $ n= 1 $ of the questioner's equation) is even finite.
So, even fixing $ n = 1 $ in the equation of the questioner one does not know how to prove that the number of solutions is finite.

Thus, the only way an answer to this question can be known, is the one I mentioned,
otherwise (and likely) this is an open problem.

I already mentioned the names Erdős and Moser; searching for 'Erdős--Moser type equation' will yield various investigations on this and related problems.

More specifically, the book 'Unsolved Problems in Number Theory' by Guy, already mentioned by Gerry Meyerson, contains a section on it (namely D7), you might be able to see it on Google books, with various references, in particular to computational work excluding solutions for $ m $ up to $ 10^{10^6} $ and even beyond that.

For more recent work on variants of this problem, see for example, a preprint by MacMillan and Sondow or a paper by Lengyel (scroll to A41).