Yes, your intuition is basically right.
It's useful to look at a few examples.
First, for a finite number of eigenstates in the sum, $C(t)$ will inevitably be periodic. Nevertheless, one still generically expects $C(t)$ to decrease from $1$ to some small value, before being "dragged back" to $1$ by the requirement that $C(t)$ be periodic. As an example, we can consider the situation $E_a = a \hbar \omega$, with the sum going from $a=0$ to $a=N-1$, and $c_a=1/N$ for all $a$. Then we have
\begin{equation}
C(t) = \sum_{a=0}^{N-1} \frac{e^{i a \omega t}}{N}
= \frac{1}{N} \frac{1 - e^{i N \omega t}}{1-e^{i \omega t}}
= \frac{e^{i (N-1) \omega t /2}}{N} \frac{\sin\left(\frac{N\omega t}{2}\right)}{\sin\left(\frac{\omega t}{2}\right)}
\end{equation}
First, note that in the limit $t\rightarrow 0$, $C(t)$ goes to $1$, as expected. Additionally, $C(t)$ is periodic with period $2\pi/\omega$, which is the period of the slowest (non-constant) mode. (You might think the right hand side is periodic with period $4\pi/\omega$ based on the sine wave in the denominator, but note that when $t=2\pi/\omega$ the numerator and denominator are both $-1$ for any $N$, and so the final result is $+1$; this agrees with the translational symmetry that is obvious in the mode expansion for $C(t)$).
Nevertheless, there are places where $C(t)$ vanishes. If we just consider the amplitude $|C(t)| \propto \left|\frac{\sin(N \omega t/2)}{\sin(\omega t/2)}\right|$, then the amplitude will be largest when the numerator and denominator are both small and have approximately the same size (near $t \approx 0, 2\pi/\omega, 4\pi/\omega, ...$), and smallest when the denominator is "large" (the most extreme cases being $t\approx \pi/\omega, 3\pi/\omega, ... $. Here is a plot showing the behavior of $|C(t)|$ for $N=\{3,10,100\}$
There are also a few interesting limiting cases to consider.
In the limit $N\rightarrow \infty$, the sum will become a Dirac comb with spacing $2\pi/\omega$. This is a extreme case where $|C(t)|$ is only non-zero when $t=0$, then immediately decoheres to zero, before being "forced back" by periodicity and repeating the cycle.
If you take the continuum limit so $a$ is a continuous variable and not a discrete index over a finite range of frequencies from $\omega=0$ to $\omega=\Omega$ (this amounts to sending $\omega\rightarrow 0$ but sending $N\rightarrow \infty$ in such a way that we keep the maximum frequency $\Omega=N\omega$ finite), then you will find that $|C(t)|$ is given by a sinc function. In other words, $|C(t)|=1$ for $t=0$, and then falls off to zero over some time scale given by one divided by the bandwidth, $\sim 1/\Omega$.
If you then take the limit $\Omega\rightarrow \infty$, then $|C(t)|$ will simply be a delta function at $t=0$. It will be non-zero exactly when $t=0$, then decohere immediately and never be zero again.
