This year I will graduate from a European university and willing to start a Ph.D. program in the USA. As a graduate student, there is a “guaranteed financing” opportunity of approximately $ 32,000 per year. My question is when a graduate student is part of a research grant (ie NIH / NSF grants) of his PI -in their own lab-, does he receive payment in addition to the guaranteed funding stipend? Does income from the research grant -if some- and guaranteed funding stipend different things?
In my experience, “guaranteed funding” means that (depending on field) you will be either working for a PI who has grant money to pay you or working as a TA (with the possible exception of your first year, where their may be general department support while you find an advisor). Because graduate stipends are typically on a pay scale, working on a grant can be expected to bring you up to the guaranteed level, but not put you above it.
To support the guarantee of funding, the university (or a unit within the university) may have a reserve fund with money that can be used to cover gaps in PI funding (which may come with TA obligations).
It varies, so you need to check with the program. For example, if the grad students are unionized at that institution, the CBA might specify that the PI can't sweeten a grad student's base stipend out of a grant. There might also be distinctions between what's permissible during the academic year and what's permissible during the summer.
I doubt that it would happen. You don't say what the stipend is for. If it is an RA then you are already being paid for research help and it might actually come out of any grants. If it is a TA then you will have certain (teaching related) duties for that and the research project is probably more associated with your own degree than as an "assistant".
But, I suppose it could happen, but don't expect a doubling of the basic grant or anything close to it. Maybe some small increment over the "guaranteed" part. The institution has to serve a lot of students and the funds aren't unlimited.
And "free money" in US universities is pretty rare.