I was thinking as follows: At each step, a PDA can put arbitrary many symbols onto the stack. But this number is constant for every individual PDA, so it can't be more than, say, $k$ symbols per step. Using only regular transitions, the stack can rise to maximally (more or less) $kn$ symbols in a run on an input sized $n$.
But what about $\epsilon$-transitions? Is there a simple argument why their maximum number should as well be independent of the input size?
So, in short: Is a PDA's stack size linear in the input size?