If you ramp way too much current through one (like 3A+), you're likely to have fused one of the bond wires open. If you examine the emitter very carefully (easiest on a high-dome) you may be able to make out a break in the bond wire and/or discoloration/burning colors on the bond wires.
I supercooled a 1W white HD, then ramped up the current to see what the limiting factor for current was. After the luxeon gave up the ghost, I examined the die and bond wires to find that the bond wire had fused open (with quite interesting remains).
Here's a picture:
the bubbled and dark blotches around the bond wire were not present before the experiment. When the wire fused open, it likely caused some arcing, and possibly boiling of the gold wire (hence the dark blotches and bubbles).