I thought it was known they can't, but it turns out it's a fan assumption.
I think the main reason it's believed is because in Movie 7 Goku had to absorb the Genki Dama due to having an impure heart and we know a pure heart is needed to ride a Nimbus. Sticking to the manga, Goku stayed Base when making the Genki Dama versus Kid Buu, even though it would make sense to stay in SSJ1 if possible, since he'd be slightly less vulnerable and MSSJ shouldn't make him any more tired than being in Base would, so he went to Base because he thought SSJ might be too impure to make and reliably hold a Genki Dama.
Plus Krillin can't ride the Nimbus but can rebound the Genki Dama. So the Nimbus might have stricter purity requirements than the Genki Dama, or it could be that they require different types of purity. If the Nimbus has stricter requirements, then SSJ Goku not being able to make a GD means he can't ride the Nimbus.