After averaging one flat every 3 years for most of my life, the last two years I had a rash of flats. One attributable to a nail, but the rest with mystery causes. After replacing one tire and both tubes, they stopped. (Also, while replacing the tire I also picked up a new bike...but that's just for fair weather use, so the old one still gets ridden.)

If I'm replacing a tire, I'll almost certainly replace the tube at the same time. Tubes cost next to nothing, so why ride a patched tube in an expensive new tire?