This clearly depends on what you mean by "provably."

Under one reasonable interpretation, the answer is very much "yes." The set $S_T$ of theorems of a (consistent, recursively axiomatizable, extending $PA$) theory $T$ is r.e., but $T$ cannot prove that the complement of $S_T$ is nonempty (Goedel's Theorem), much less that any specific element is in the complement of $S_T$. $\Box$

I think this probably addresses the question you ask. However, this might be unsatisfying, since a theory slightly stronger than $T$ - namely $T+Con(T)$ - can prove that certain sentences are not theorems of $T$. Maybe we don't care about "provability from $T$," but rather "provability from $T$ plus reasonable consistency assumptions." So now, we might ask something along the lines of:

Fix a theory $T$ which is a consistent, recursively axiomatizable extension of $PA$ (and, for simplicity, a subtheory of true arithmetic). Let $T^\alpha$, for $\alpha$ an ordinal, be defined by: $T^0=T$, $T^{\alpha+1}=T^\alpha+Con(T^\alpha)$, and $T^\lambda=\bigcup_{\alpha<\lambda} T^\alpha$ for $\lambda$ a limit. Is there some co-r.e. set $X$ which has no element provably in $X$ in any of the $T^\alpha$?

A stronger question:

Does some "ordinal iteration" of $T$ prove every true $\Pi^0_1$ sentence?

If the answer to this question were "yes," then this would be a sense in which the answer to your original question is - morally, at least - "no."

There's a huge problem here, however: for $\alpha\ge\omega$, $T^\alpha$ is not uniquely defined by what we've written. To make this idea precise, we need to use *ordinal notations* - that is, ways of representing ordinals by natural numbers - and now things get very messy. Alan Turing wrote a paper about this in 1939, in which he proved:

Fix a true $\Pi^0_1$ sentence $\varphi$. Then there is some notation for $\omega+1$ according to which $T^{\omega+1}$ proves $\varphi$. More precisely, we define $T^a$ for $a$ a *notation for an ordinal*; there is then a map $\vert\cdot\vert$ from notations to ordinals - the interpretation map - and Turing's result is that there is some $a$ with $\vert a\vert=\omega+1$ such that $T^a$ proves $\varphi$.

This isn't the end of the story; we can look at "nice" systems of notations, and try to avoid Turing's result. But things stay pretty complicated, still. I don't know much about this, but Feferman and Spector's paper "Incompleteness along paths in progressions of theories" and Franzen's book "Inexhaustibility" both go into more detail.

EDIT: Another resource you should look at is Lindstrom's book "Aspects of Incompleteness" (see Andres' comment below). (Totally unrelated, but I can't resist plugging it: Lindstrom proved one of my favorite theorems in logic - that first-order logic is the "strongest logic" with the compactness and Lowenheim-Skolem properties.)