It is a consequence of total boundedness of bounded intervals in $\mathbb{R}$ that uniformly continuous functions on such intervals are bounded.
What is the best example of an unbounded uniformly continuous function on a bounded metric space? I suppose the open unit ball of $\ell_2$ should be a good place to start.
David Mitra pointed out an example of an unbounded uniformly continuous function on a bounded metric space. But to answer the question posed in the title, I will show that every uniformly continuous function $f$ on the unit ball $B$ of a normed space is bounded.
Indeed, uniform continuity implies the existence of $\delta>0$ such that $|f(x)-f(y)|\le 1$ when $|x-y|\le \delta$. Pick $n>1/\delta$. For any $x\in B$, $$ f(x) = f(0) + \sum_{k=1}^n (f(kx/n)-f((k-1)x/n)) $$ where every term of the sum is at most $1$ in absolute value. Hence $$|f(x)|\le f(0)+n$$ proving that $f$ is bounded.
More generally: let $E$ be a metric space. Suppose there is a constant $L$ such that any two points of $E$ can be joined by a curve of length at most $L$. Then every uniformly continuous function on $E$ is bounded. The argument is essentially the same: use a suitable partition of the curve.