So far, I've never seen an original or a renewal where the papers couldn't be turned in by the last day. Of course, having a plane ticket only a few days after the 30 day deadline will make this a real nail biter.
The trick (that I have learned by serving as a bad example of what NOT to do) is to jump on it on June 1st (or 2nd). If a medical is needed, this should get it back to you by Tuesday, June 7th at the latest. Also, grab a Chinese friend who can translate (unless your Mandarin is really good), and volunteer to carry your own paperwork around town.
NOTE: If you have multiple PSB stations to visit (currently, I have to go to 3 before I can go to the central one), all the lesser stations don't care about the medical. If a medical is needed (it may not be in your case - rules on this seem a bit variable), it will only be needed by the final station that issues the Res Permit. This means that you can start any sub-station procedures, labor bureau procedures, etc. while waiting for the medical results.