When a graduate student be a part of a research grant, does he get paid in addition to a university stipend? 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).