Firstly, the post office isn't closed for the final two weeks of December that I'm aware of. If your plaintiff is making this claim, you should demand proof from his local post office that it was closed during the time he claimed. How did you mail the interrogatories? As with all legal correspondence sent via US mail, I recommend sending certified mail with return receipt requested.
It is, however, fair to assume that he could not have picked up the interrogatories on 12/18, the day you mailed them. If the letter was not delivered, but held at the post office, then the post office will have a record date as to when they first attempted delivery. That would also be the first day your plaintiff would be on notice that he had mail to pick up, by the way. The issue should be fairly fact-specific.
While there isn't a time limit per se, the court will hear your argument that the plaintiff neglected to pick up the discovery package in bad faith, and he should not get the benefit of extra time because he unreasonably delayed.
Good luck: if you have any further questions, please feel free to contact my office.
The foregoing is not legal advice, and nothing in the foregoing shall be deemed to create an attorney client relationship. If you feel you need to speak with an attorney regarding your issue, it is recommended that you contact an attorney with expertise in your area of inquiry. The information related above is purely for informational purposes, and should not be acted upon without speaking with qualified counsel familiar with you specific situation and the laws related thereto.
The specific rule is that service of interrogatories is deemed complete on the day they are mailed via ordinary mail. I believe the court rule is 1:5-4. The procedure when a party fails to respond to discovery on time is not affected by whatever explanation they offer about the delay. You can file a motion to dismiss, the form is here: http://www.judiciary.state.nj.us/prose/10915.pdf. Good luck.
I agree with the other answers provided to you. If you want to go to trial (perhaps because you have a counterclaim or some other reason), you can send a letter to the court clerk advising that you do not have answers to your interrogatories and therefore you want the trial date to be adjourned to allow time for you to get the interrogatory answers. And file a motion to compel the plaintiff to answer your interrogatories. Or simply move to dismiss, as another attorney has suggested. Just don't wait until the trial date to take action - judges don't like that.