Oh, absolutely! As long as you have a library card for that library (and they are supported by Overdrive), you can access as many libraries as you wish. I have library cards for six different libraries and sometimes have books downloaded from several different ones. Helps when you start seeing the long waiting lists...
Overdrive keeps track of the books for you (returns and such) and it doesn't matter from which library the book is - it's there on your shelf for reading when you get to it.
Marilyn