They were right... but it seemed to be a fairly new issue and I never could figure out why. It turns out it all comes back to 32766.
Apparently, Opera can't handle widths greater than 32766px. There is a thread in Opera's forums which means I'm not alone here.
Reader Erdei Csaba clued me in:
32766px ... is the highest signed 16-bit number (32767) - 1.
Opera must store these values as signed 16-bit numbers. And apparently higher positive values (> 32767) are processed as negative values. The CSS1 spec explicitly forbade negative width values, but the newer specs don't really say anything about it. They probably assume they didn't have to say anything, because it's obviously pretty ridiculous to have a negative width.
Long story short: the maximum width of an element in Opera is 32766px, and that probably should be fixed.