According to the specialists,

http://www.straightdope.com/classics/a5_266.html

"....Lewis Carroll himself got bugged about this so much that he was moved to write the following in the preface to the 1896 edition of his book:

Enquiries have been so often addressed to me, as to whether any answer to the Hatter's Riddle can be imagined, that I may as well put on record here what seems to me to be a fairly appropriate answer, viz: `Because it can produce a few notes, tho they are very flat; and it is never put with the wrong end in front!' This, however, is merely an afterthought; the Riddle, as originally invented, had no answer at all.

Did this discourage people? No. They figured, that dope Carroll, he's too dumb to figure out his own riddle, setting aside the halfhearted attempt just quoted. So they ventured answers of their own, some of the more notable of which are recorded in Martin Gardner's The Annotated Alice and More Annotated Alice:

• Because the notes for which they are noted are not noted for being musical notes. (Puzzle maven Sam Loyd, 1914)
• Because Poe wrote on both. (Loyd again)
• Because there is a B in both and an N in neither. (Get it? Aldous Huxley, 1928)
• Because it slopes with a flap. (Cyril Pearson, undated)

Not bad for amateurs. But the real answer, to which the careers of Poe and Carroll bear ample testimony, is that you can baffle the billions with both.