DME is slant range, ATC may be horizontal. How high were you?

I was at 2000 feet, which is about 1700 AGL.

Certainly not enough to give you an extra mile of slant range.

You have a digital readout that gives exact distance. The controller
looks at the little space between you and the VOR symbol on the radar
scope. That symbol, a circle, is about a mile in diameter. The
controller looks at that space and he came up with 3 miles. He was off
a mile. No big deal.

Another factor is the physical location of the DME site with respect to
the airport reference point.