Unlikely but possible. Nothing you should need to think about.

Every write operation on the card is risky and could trigger corruption if something nasty happens exactly in the moment of the write operation. Deleting a photo is a write operation. The more of those happen, the higher is the risk. Obviously moving the deletion operations AFTER a backup minimizes the risk of data loss. You still have the image saving write operations but are on the safe side with the deletion write operations. So technically the advice is valid.

But the base risk is so minimal that it doesn't matter. The cards (at least common SD cards) are quasi self-contained; the operation works or it doesn't. If it doesn't then the whole card is broken and will probably fail at the normal image saving write operation; don't forget that the only avoidable extra risk is in the deletion operations, a tiny fraction of the operations you do on the card. You shouldn't use the cheapest cards or cards that already showed flaws. With a good quality card the base risk of write operations to fail is so minimal that the additional risk of the deletion operation statistically will not realize. I can't put it better than this XKCD does: http://xkcd.com/1252/

The problem is to argue with someone who insists on the correctness of this advice. It is technically correct, even if borderline insignificant, but the suggested action is not useful. Social advice here: If they want to, let them. It doesn't harm anybody, it doesn't harm them. The "only" negative outcome would be a photo not taken because of a full card. This happens more often than the technical failure on deletion. The negative outcome outweighs the benefits so you shouldn't follow the advice yourself.