Oh yeah. Careful about mentioning the school and make sure they are cool with it. I have overheard some nurse managers complain that they do not like nursing students as CNAs. That is because nursing students (according to them) are more likely to take days off and the closer they get to graduation, tend to not care about the job. They can not be threatened with unemployment or motivated either as easily. They come in tired and do not bust tail as hard. Most do not give two flips about your personal goals and only care what you can do for them as a CNA, though there are exceptions.
You do know some 4 year RN schools have paid internships? One person I knew in this area got a sweet tech gig at a pediatric hospital.