Yes, I would interpret these differently.
You can possibly do this.
The difference is in the emphasis, what the reader focuses on. This sentence emphasizes "you can". Therefore you are saying "you can do this", with a subtle caveat of "possibly". The "possibly" is subtle, a slight doubt introduced to indicate not to be 100% sure. You can do this, I'm 90% sure.
You possibly can do this.
This sentence emphasizes "you possibly". You are saying yes, this is possible but unlikely. You perhaps, if the conditions are right, might be able to do this but it is highly unlikely.
Both sentences basically mean the same thing, but for native English there is a very subtle difference between them. This is indicated in spoken English by inflection. I would say " you POSSIBLY can do this" to emphasize the significant word.