Can students apply for Research Grants (US)? I know you can apply for fellowships for individual students, but I'm not talking about funding only for one's self. Let's say I noticed a grant that was solicited on grants.gov and I wanted to apply for it. Can I do that?
The nature of these grants is to supply the awardee with funding for experimental set-ups, student funding (undergrad/grad), and other various research supplies. 

&amp; As a follow up question, is it possible to apply, if your PI is applying for this funding as well. Basically, could I send in a proposal and my PI could as well? Are there some restrictions here? I've read through the solicitation's eligibility and couldn't find anything about multiple names being on different proposals, or if we were to apply separately, then having the funding go to the same group under different names. 
Just curious if anyone knew.
I'll write back, if I find more info.
Technically, in the US, most grants are not normally submitted directly by the principal investigator, but are instead sent in via a "grants office" that acts as the interface between the principal investigator and the funding agency.

Consequently, in addition to the grant-specific funding rules you have from a specific funding agency, you also have to deal with whatever restrictions your university grant-office has. For instance, it's not likely that your grants office will accept a proposal submitted by a graduate student instead of by a faculty member or similar personnel (lecturer, research professor, etc.).

In general, though, you need to read the requirements of the individual granting agency, as well as ask your local grants office.