My 2003 Accord was originally purchased in August 2003. Right now it has a little over 35,500 miles.
Our old '82 Accord had at least 200,000 miles on it. Probably more. It was my mom's, then it was my grandmother's. My grandmother got rid of it because of an accident, it would have cost too much to repair it. It still ran like a champ, though.
My mom's 2002 Accord has over 45,000 miles on it.

We have never had any problems with the engines of our Honda's. We've had a first gen Civic, first gen Accord, '82 Accord, '90 Accord, '97 Odyssey, '02 Accord, '03 Accord. We've also had numerous Honda motorcycles.