The Staffords

The Tudors had a fragile claim to the English throne - there were others that could plausibly have a far stronger claim to the crown. Some of these claimants were just one step away in the shadows of the Royal court, anticipating, patiently waiting for their own chance to shine. This included the Stafford family.
Humphrey Stafford, the first Duke of Buckingham was a nobleman that fought in the Hundred Years War and in the War of the Roses. Through his mother, Anne of Gloucester, he was a descendant of King Edward III. He was to stay close to the throne by fighting onside with King Henry V and became a councillor to his heir, King Henry VI. Humphrey's grandson, Henry Stafford, second Duke of Buckingham, also stayed close to the crown. He was also one of the main suspects for the purported murder of the 'Princes in the Tower' and would eventually be executed for treason by Richard III.
Henry Stafford's son with Katherine Woodville, Edward Stafford, third Duke of Buckingham would eventually meet the same fate as his father, however this time it wouldn't be on the orders of the Plantagenet King Richard, but on the order of the Tudor King Henry VIII instead.
Connections to famous noble families such as the Nevilles, the Poles and the Howards were plentiful, but circumstances and fate always seemed to intervene. The Staffords remained close, but never quite managed to sit on the English throne themselves.