We show that the spectral measure of any non-commutative polynomial of a
non-commutative $n$-tuple cannot have atoms if the free entropy dimension of
that $n$-tuple is $n$ (see also work of Mai, Speicher, and Weber). Under
stronger assumptions on the $n$-tuple, we prove that the spectral measure is
not singular, and measures of intervals surrounding any point may not decay
slower than polynomially as a function of the interval's length.