It sounds very similar to Bio with the exception that the department typically has money for their students coming in(the best schools have training grants from NIH which cover incoming students). Tuition is always covered and is only up to the PI to pay when the student's departmental money runs out, but in practice, the dept. picks it up if the student TAs.
At good schools, students are encouraged (and in some cases, mandated) to submit a grant to NIH which covers their tuition/stipend and provides some travel money for conferences. That grant is really difficult to get, so most 3-6th year students are covered under their PIs grant, or they teach to cover tuition.
ALso, and I think this may apply to CS people, there are dissertation grants which give a small amount of cash (when I applied it was ~$15k) to improve a dissertation project.
NSF has some really nice programs out there. I've been funded by them on 2 different occasions and I really like what they support.