Or is it usually not possible?
Does this also depend from department to department? I'm in a geoscience department, and I was recently advised that due to a limited number of positions, students can only expect to be TAs for 1/4 of their time here. Geoscience does tend to have significantly fewer courses than most other fields though, so I wonder if it varies from field to field?
Does it also vary depending on whether the school is public or private?
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.
This does vary from school to school and field to field, and it would be difficult to classify public or private schools as being in one camp or another. I have known people who never TA'd and I've known people who have TA'd for their whole graduate career. Here are some plusses and minuses (and certainly my own opinion):
Plusses to only TA'ing:
You spend more time teaching, although some TAs spend all their time grading, which can be tedious and not particularly beneficial to you.
If your advisor isn't paying you from his/her grants, you could have more leeway for a more self-directed PhD. Obviously, your advisor still has a large roll to play, but ultimately less control if you want to push the issue. This is generally more true if your funding comes from an external fellowship.
If you're in a field where grant money is limited, this is a good way to get funding for your degree.
Minuses:
TA'ing takes up time that could be spent on your own research. The point of Research Assistant fellowships is explicitly for you to do research.
As a corollary to #2 above, your advisor might not feel as much ownership of you, and you might find that he/she isn't as willing to spend the time to help you with your research.
Other students might wonder why you can't find research funding.
I know students who love being TAs, and they actively try to find new TA positions because of that and not because of the money.
Is it possible for a student to TA their entire way through graduate school?
Yes, it is possible.
I did it. I'm betting most academics can list someone they know who did it for whatever the reason might be. I did it partly because I enjoyed it and partly because the grant money was tight. I was able to do it because there were not enough first year graduate students to fill every open TA position. I suspect that the situations the lead to someone teaching their way through grad school have some features in common with my situation.
You don't say so, but the tone of your question implies that you might want to do this. If so, ask the appropriate person who controls such things - the chair, assistant chair, undergraduate coordinator, or even the professor of the class you want to TA for. If you are good at it they might let you with your thesis adviser's support.
The extra teaching experience will be beneficial if you intend a career at a 4-year liberal arts school.
However, if there are a limited number of TA positions, those will go to the first year graduate students because those students 1) are not likely to be supported by grants or fellowships, and 2) probably have a TA requirement to complete the program.
