The Principality of Monaco has a royal family which has also endured its share of suffering. Princess Grace, actually an American actress who had a fairytale romance with her husband Prince Ranier, died young in a tragic car crash on a treacherous road--and her daughter was involved. From there, tragedy has followed this family, though they live in a beautiful place. It's hard to know if nobility has the same problems the rest of us have, only bigger; or if they really do have more troubles. It's impossible for them to live privately, good or bad, that's for sure.
I suppose this answer may depend upon your own definition of "suffering." A more recent example may be that of the present British royal family. The late Princess Diana had to undergo the negative scrutiny of Queen Elizabeth as well as the marital indiscretions of her husband, Prince Charles. Her counterpart, Sarah Ferguson, wife of Prince Andrew, was never accepted wholly by the royal family, and her marriage ended in divorce. She has had problems restricting her extravagant lifestyle and recently was further disgraced when she was filmed during a sting operation accepting a bribe. Prince Charles' personal life has not always been a happy one: His seemingly idyllic marriage to Diana ended with the public siding with Diana, and he has been obviously bypassed by his mother, Queen Elizabeth, for consideration of becoming king. His own personal life has been darkened further by the death of Diana, whose own popularity grew even greater following her tragic accident. His present female interest, Camilla Parker Bowles, has been trashed in the press and scorned by Charles' parents.
King Henry the VIII of England's first wife, Queen Catherine, had a rough life! She was a true princess, daughter of Queen Isabella and King Ferdinand of Spain. She was first married to Henry's other brother, Authur, who died within weeks of their marriage. To maintain herself and the alliance between England and Spain, she then set herself up to marry Henry. She claimed that due to Authur's illness, they never consumated their marriage, therefore it was never a real marriage, and that she was completely in the right to marry Henry.
Once they married, they had a daughter, Princess Mary, but the Queen was never able to carry a live baby boy to term, and Henry was desperate for a male heir. He grew dissatisfied with their marriage and he fell in love with Anne Boleyn. Henry spent several years trying to get an annulment or a divorce from Catherine, but she refused to give in on any grounds. He tried to claim she lied about the consummation of her marriage to Arthur, thereby rendering the second marriage invalid. She swore up to the day of her death that she was a virgin for Henry. He tried to get the Catholic pope to grant him a divorce, but when all attempts failed, he disassociated all of England from the Catholic Church, named himself the head of the new Church of England, and essentially granted himself the divorce he so desperately wanted. Queen Catherine was steady in her emotional and spiritual battle with Henry until the day she died in a outcast state of neglect with only a very few loyal companions with her. Henry made her life miserable.
This answer could gone on to retell how 2 of Henry's wives were beheaded on his command (some significant suffering there!)