What the title said.  In a slightly more leisurely fashion:-

> Let $X$ be a compact, connected subset of $\mathbb{R}^2$ and let $x\in X$.  Can $X\smallsetminus\{x}$ be totally disconnected?

Note that the [Knaster--Kuratowski fan][1] shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.

To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.

  [1]: http://en.wikipedia.org/wiki/Knaster%25E2%2580%2593Kuratowski_fan