Is it possible for a student to TA their entire way through graduate school?
The ability of a department to fund TAs depends on the courses being taught, the number of students in a class, and the overall budget. Often, a department will use this number in combination with the amount of funding faculty are pulling in as well as other sources to decide how many students to admit and how much TA funding support to provide. 

Independently, a department might make a determination of how long any one student might be guaranteed a TA position (this is often detailed on the admissions letter), with the expectation that students will look to be supported by research funding, or are expected to support themselves. Again, this depends from department to department and from university to university.

I don't think there's any area-wide standard. In computer science, I see all kinds of models, ranging from full support through the entire program to no funding at all (although this latter model is rare). 

So to answer your question: maybe, it depends.