I understood that you were granted 3 days to allow for forgetting what date your subscription expires (I have no idea on mine) or payment problems.

I know some players have had success petitioning when their training expired before these 3 days were up.

Edit:

I would also add that if your account is banned then it matters not if your training continues. I expect if you convinced CCP that the ban was not in order and they reinstated the account then you could get your training time back again.

Additional info in response to duplicated question and comment to this post:

OK I am now pretty sure that there is nothing official stating 3 days in the whole Eve website having spent some time trying to find it.

Werner's comment to this answer shows that training definitely continues for at least nearly 3 days and there are many more posts in the forums stating 3 days than any other firm figure. Of course this may have all been started by one person guessing but I feel the evidence is more than strong enough to be sure.

There are numerous mentions, mostly about "friends of friends", who reported that skills with more than 3 days left didn't train at all but this was known and fixed as a bug by CCP who reinstated lost training when petitioned.

There is also some anecdotal evidence that only the skill that is actually training when the account expires continues to run so it doesn't appear to continue through your skill queue for 3 days. It's possible that these last two points were related and that therefore this may also no longer be true.