Yes, you can use the family permit up to its last day of validity as long as your wife is in the UK or traveling with you when you enter.
The EEA family permit is different from a visa in that it does not authorize you to be in the UK. Rather, your right to be in the UK flows directly from EU law and the UK immigration regulations that implement that law. So once you're in the UK, it's fine for the permit to expire, and border officers should not seek to establish that you will leave before it does.
The government's pages on the family permit say as much:
Stay after your EEA family permit expires
You can stay in the UK after your permit expires if you:
Note how the link for eligibility to stay after the permit's expiration points to the page that describes eligibility for the permit itself.