If $a_n(t) \to a(t)$ and $t \mapsto a_n$ is continuous, are the $a_j(t)$ in a compact subset independent of $t$?

Question

I have a sequence $a_n(t) \to a(t)$ in a Hilbert space $H$. Furthermore, for each $n$, $a_n:[0,1] \to H$ is continuous. Is it possible to deduce that $$\{a_n(t)\}_{n \in \mathbb{N}}$$ is contained in a compact subset $K$ of $X$, where $K$ is independent of $t$ as well as $n$???

I think the asnwer is no, since we only get inclusion in compact sets that either depend on $t$ or on $n$. Does anyone know if it holds under additional assumptions on the sequence?