Sure a sequence of continuous functions converging pointwise to a continuous function do not necessarily converge uniformly. A good characterisation of this behaviour is that such a sequence converges quasi-uniformily (Arzelà–Aleksandrov theorem).
Does quasi-uniform convergence implies almost uniform convergence ?
Here's the definition of quasi-uniform convergence
A sequence of mappings $f_{n}$ from a topological space $X$ into a metric space $Y$ converging pointwise to a mapping $f$ is called quasi-uniformly convergent if for any $\varepsilon>0$ and any positive integer $N$ there exist a countable open covering $\{\Gamma_{0},\Gamma_{1},…\}$ of X and a sequence $n_{0},n_{1},…$ of positive integers greater than $N$ such that $|f(x)-f_{n_{k}}(x)|<ϵ$ for every $x\in\Gamma_{k}$