No, you can't do both at the same time. We've had associates at our store think they could (after we repeatedly told them) and they had to pay the STD company back. I think the associates think we are lying to them, but we can't convince them otherwise.
it is confusing though. we had an associate go to our market HR person and she told them they had to use all of their vac/sick/personal time first and then they could collect STD. Not correct.