The one thing that really cuts you off from filing an answer after your 35 days is not the time frame of thirty-five days; it is the request by opposing counsel to find you in default. This filing is what really controls the filing of an answer.  However, even when you have been defaulted, it still may be possible to file a late answer after having the default vacated.
As an example, let's say that after the 35 days are up, it is now day 45 maybe you were or weren't in the New Jersey Foreclosure Mediation program. However the bank's attorney has not filed to have you defaulted. If you file an answer the court will most likely accept it as if it was filed on time. I've done this on numerous occasions when clients had not even hired me until after the 35 days had run out.