Usually the fuzziness comes from one group paying for research that another group is doing.
Grad students and post docs work in government labs, like National Institute of Standards and Technology - NIST or the Naval Research Laboratory.
Academic labs work on corporate grants - you can see this in everything from subcontracting on government projects to the origins of Genentech. Genentech was started by bringing together several pieces of existing research and subcontracting to university labs
Government labs like National Institute of Standards and Technology - NIST work on corporate projects through grants the same way they work for the government.
Corporate R&D works on Government grants all the time.
In this whole structure, People giving out grants try to derisk them as much as possible so it is more likely that a more ‘established’ institution will get the grants.