Many people seem to know the following. Personally, I don't quite understand it though and maybe I'm wrong. It's the fact that "a vector bundle on an open subscheme extends in only one way to a vector bundle on the entire scheme" (under some dimension hypothesis). Let me be more precise.

By a vector bundle I will actually mean a coherent locally free sheaf on $X$. A scheme will be noetherian and connected.

Let $U$ be an open subscheme of a scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $F$ be a vector bundle on $X$ extending $E$, i.e., $F|_{U} = E$. Then any other extension of $E$ to $X$ is isomorphic to $F$.

I can kind of see why this goes wrong in codimension 1. The subscheme $Y$ is a divisor and these correspond to line bundles in a way. One could then just tensor an extension of $E$ with any line bundle $L$ on $X$, right?