It seems to be a standard result that the number of degree-1 representations of a group $G$ is equal to $[G : G']$ where $G'$ is the commutator subgroup (e.g. Lemma 6.2.7 in the 2012 textbook "Representation Theory of Finite Groups" by Steinberg).
A consequence is that a group is perfect (i.e. $G = G'$) iff the only degree-1 representation is the trivial one.
Is there any analogous theorem that bounds the number of degree-$(d > 1)$ representations of a perfect group, or gives some other characterization of them (that is unique to perfect groups)?
Let $G_p = \operatorname{SL}(2,p)$ for $p$ prime, $p \geq 5$. Then $G_p$ is a perfect group and has $\frac{p-1}{2}$ characters of degree $p-1$.
Taking $p$ to be the first prime greater than or equal to $2n+1$, we get that $G_p$ has more than $n$ characters of degree $p-1$.
Most groups of Lie type have high multiplicity character degrees (multiplicity above some multiple of $q^r$ where $r$ is the twisted lie rank and $q$ is the smaller of the possibilities for $q$, so that $q^t$ is the field size with $t$ the degree of the twist).
I didn't check alternating groups, but they probably work as well.