The other moment, I tried to do just this. But, epstopdf complained about the lack of a bounding box in one of my EPS files. Indeed, the conversion finished but generated a huge white background with the actual image somewhere in the lower left corner. From the man-page:

epstopdf transforms the Encapsulated PostScript file so that it is guaranteed to start at the 0,0 coordinate, and it sets a page size exactly corresponding to the BoundingBox. This means that when Ghostscript renders it, the result needs no cropping, and the PDF MediaBox is correct. The result is piped to Ghostscript and a PDF version written.

If the bounding box is not right, of course, you have problems…

Luckily, while tab-completing from eps to epstopdf, I noticed the eps2eps utility. I though: What if this utility happens to sanitize the EPS file a bit? A quick look at the man page and a test run later, my hope was confirmed: epstopdf would now generate a nice PDF file without complaining.

The epstopdf manual page could be amended to: If the bounding box is not right, you might want to try to run eps2eps first.