Well, yes and no.
Yes, it can be done. We have reprogrammed HP12c into HP15c and HP16c - with the proviso that you need to place an overlay in front of the keyboard since the key assignment is completely different!
However, to do so you need a suitable file (there are several available in the web), but, most importantly, you need to get a programming cable - and that's the difficult thing, since it is a non-standard cable. For a while HP produced some 1000 cables for the HP15c Limited Edition. Now, this cable fits in the hole for the new Voyagers - but uses a different pin scheme and protocol - the latter being USB protocol, while the old HP15c LE has a different protocol. So even if you have the cable, you need to modify it so that it can be used! There are instructions available if you have managed to secure one of the original cables.
Once you have that, you need an Atmel-provided program to download the firmware. That is an easy part, once you have the above. In fact, it is an easier process than the old calculators: you just need to press and hold erase in the cable, then press reset, then release erase - and you're set to go! I have not managed to brick a single unit - while I bricked several of the old HP15C LE units.