Can we "reject" that a distribution is a finite mixture of normals? Consider a one-dimensional distribution function $f(x)$. Suppose this distribution has all the nice properties, such as continuity, smoothness, etc. We observe $f(x)$.
Suppose that we believe that $f(x)$ can be represented as a mixture of $5$ normals each with mean and variance $(\mu_j, \sigma^2_j)$ respectively:
$$
(1)\quad f(x)=\sum_{j=1}^5 \gamma_j G(x; \mu_j, \sigma^2_j)
$$
We don't know $\{\mu_j, \sigma^2_j, \gamma_j\}_{j=1}^5$. However, we assume that $\gamma_j>0$ and $0<\sigma^2_j<\infty$ for each $j=1,...,5$.
Question: Could it be that there exists no $\{\mu_j, \sigma^2_j, \gamma_j\}_{j=1}^5$ such that $f(x)$ can be represented as in (1)? In other words, is it possible in principle to reject (1)? Note that I'm not asking for a formal test of (1). I'm just asking whether it is true or not that $f(x)$ can be always represented as in (1).
 A: No, it's not true (that you can always represent a distribution by a 5-components Gaussian mixture - the opposite statement in the question title that you can "reject" it, is true). Gaussian mixtures are identifiable (*), which means in particular that no Gaussian mixture with, say, 100 mixture components can be represented with a lower number of components, say 5. This alone already refutes your suspicion (actually it should be intuitively clear that you can't represent a distribution with 100 modes by a 5-component Gaussian mixture). I will however add that most (reasonably well behaved) distributions that are not Gaussian mixtures can be arbitrarily well approximated by Gaussian mixtures assuming that the number of mixture components is allowed to be arbitrarily large. See Approximation by finite mixtures of continuous density functions that vanish at infinity by Nguyen, Nguyen, Chamroukhi and McLachlan. If you limit the number of mixture components to five, this will not work.
Obviously a Gaussian mixture with five components has many parameters and is quite flexible. It can well approximate many distributions, and often a very large number of observations will be needed to see the difference. But it exists.
(*) Technically for this to hold one needs to forbid label switching, zero weight components, and different components with the same parameters, but that's rather irrelevant to the question. The identifiability result is in Yakowitz, S. J. & Spragins, J. D. (1968). On the identifiability of finite mixtures. Ann. Math. Statist.
39, 209–214.
