Somebody way upthread mentioned the two skeletons found near Edward IV's tomb at Windsor. This is what Wiki says:
In 1789, workmen carrying out repairs in St. George's Chapel, Windsor, rediscovered and accidentally broke into the vault of Edward IV and Queen Elizabeth Woodville, discovering in the process what appeared to be a small adjoining vault. This vault was found to contain the coffins of two mysterious, unidentified children. However, no inspection or examination was carried out and the tomb was resealed. The tomb was inscribed with the names of two of Edward IV's children: George, 1st Duke of Bedford who had died at the age of 2, and Mary of York who had died at the age of 14; both had predeceased the King.[16] During the excavation for the royal tomb house for King George III under the Wolsey tomb-house in 1810-1813 two lead coffins clearly labelled as George Plantagenet and Mary Plantagenet were discovered and moved into the adjoining vault of Edward IV's but at the time no effort was made to identify the two lead coffins already in the vault.[17]
It goes on to say this area was being worked on again in the late 1990s and a request was put in to open the vault and examine the two unknown occupants, but the royal household turned it down. I wonder if (assuming the Leicester skeleton is Richard) another request will be made again now?
I can see why this kind of thing is only done very occasionally though, the royal household has a duty of care and it's not massively respectful is it, digging up skellies to test every time we have a scientific advance! Would be a very interesting one though.