What is the most an international student could earn at a US grad school?
I can't answer your question directly (probably no one can), but I have some first and second-hand knowledge. This applies to CS/engineering in the US:


I've had several friends that went to Cornell, Princeton, and Stanford and they have told me that they make around $28,000-$34,000 for a 12 month salary. However, the cost of living in places like Stanford is very expensive, so you need to adjust for that (there are calculators online).
You can do an internship at a company or lab and make around $30,000 in a single summer. 
Most funding agencies and departments won't allow you to double dip. So receiving external funding (e.g., a fellowship) would replace, not supplement, your assistantship with your department.


In ideal circumstances, this means that you could make 25k (9 month salary) while at university and another 30k during the summer, for a grand total of $55,000 before taxes.

Of course, you have to be careful because internships aren't guaranteed and you may have to relocate to another city/state for them (often paid for by the company) which might pose another challenge for your family.