(I'm not adressing here the issues of the "specific person", it is I believed addressed well enough by other answers, whether electronically or in the real world)
You are here dealing with notarization. You need a "legally trusted" 3rd party, that is a 3rd party whose statement will be accepted by a court in the jurisdiction(s) you are interested in.
When you say "at a specific moment", it's a bit of tricky question. Like Ja1024 said, using a 3rd party service like Dropbox might do the job for you, you would need to talk to a lawyer.
Now I'm wondering if your question is properly asked.
You could want to prove that you got a file "before a specific moment". That's the easy one. A lot of notarization services could do the job (typically with hashes). Or you can go for the the "all out" solution: in some jurisdictions, you could leave a copy of the files to a legal officer who will keep your files in a safe and give you a certified document they were deposited at such date. In others, leaving the document with your lawyer might be enouh. Using the hash solutions, I know jurisdictions where you could write the hash of the file (and size) on a paper, and just get a doctor to sign that you presented them the paper at a specific date.
If you want to proof that you got the file at a specific time, meaning "not before, not after", that's the above, plus proving you DIDN'T get the files BEFORE a specific moment, or in another words you got them "after a given moment", and this is virtually impossible. That would mean having someone certify that at no point did you get those files before, which basically implies monitoring your every interaction prior to that moment...