Short answer is "usually, yes." Slightly longer is "depends on the program." And, always, caveat emptor.
Many of the hard science programs require their incoming grad students to take a battery of what are essentially placement exams to figure out where they need to "top up" before continuing on with advanced coursework. This is easily demonstrated in chemistry programs where many schools don't offer much in the way of inorganic chemistry, or don't have extensive analytical courses. Students that don't place out of inorganic frequently have to take a 400 level undergrad class.
Done in this manner, the courses are part of the degree program, and are covered by the normal financial aid package.
If you are looking at taking classes (i.e. prerequisite courses for med school apps) that are not part of your degree program, then you will almost certainly have to work something out with your university where you pay for your 'personal' classes that are independent of the degree program you are enrolled in. That said, the school will almost certainly not care about what classes you are taking, so long as it doesn't interfere with your degree progress.