Lots of screwups and inconsistencies in the movie, but regardless, it's still a classic.
I remember someone telling me a long time ago that they noticed that the Charger lost three hubcaps during the chase scene... all off the left side. The next time I saw the movie, I counted and I think I remember seeing all three.
Bullitt's Goofs section at IMDb.com has probably every screwup here:
http://www.imdb.com/title/tt0062765/goofs
Here's a very interesting one:
During the chase scene you can hear a manual stick shift changing gears on the Mustang during exterior shots of the car, but shots inside the vehicle during the chase show an automatic and not a manual stick.