Can router bits be sharpened?

[Timberwolf]

What John said.  I also sharpen my forstner bits as well.  One thing I do is only sharpen the inside edges and not the outside and you do not have to press hard on the diamond hones or you will lose the abrasive Yes I did it.
  I also do a brief sharpening when I am done every time as well to keep them sharp.

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Yep...I just sharpen the face of the cutter with a diamond hone....not the outer edge..

I could probably sharpen one to some degree on the outer edge using a diamond cylinder burr in a Dremel, but that would reduce the diameter more.