Well Dave, you could potentially set up some cool thingy where you have
1) a file input and
2) a text input positioned directly over the file input.
You could have the browse button visible, the text input over the file input's text area. On the file field's onchange event, you could parse the file name out with JavaScript (something like String.split("/")) and then populate the text box with the file name.
I didn't feel like looking into the large amounts of CSS and x-browser functionality, however, since I'm lazy. Now that I've brought this up, though, I may need to.
*cLFlaVA
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Lois: "Peter, you're drunk!"
Peter: "I'm not drunk, I'm just exhausted from stayin' up all night drinking!