Anne may not have been royalty but she was highly connected, and aristocratic maternally. Her father may have only been made an Earl after Anne hooked up with the king, but his father was a knight of the bath by Richard III and high sheriff, and his grandfather was Lord mayor of London. His mother was Lady Margaret Butler, the daughter of the Earl of Ormond. She was co-heiress to her father's fortune, outside of Ireland where a male cousin inherited, and the land and money Mary Boleyn and her husband William inherited after the death of her parents, came from her. Of course Anne's mother was the daughter of the Duke of Norfolk.
Henry VIII may not have married princesses as was expected, with the exception of Katherine of Aragon and Anne of Cleves, but his wives were all well-connected and most at least partially aristocratic (maternal side). Katherine Howard was of course Anne's cousin, and artistocratic on her dad's side. Edmund Howard was very poor (Elizabeth Boleyn's brother) and left her in the care of his stepmother, the dowager duchess of Norfolk. Jane Seymour is actually related to them as well. Her father was gentry, knighted for service to Henry VII an VIII but her mother was first cousins with Elizabeth Boleyn and Edmund Howard. Margery, Elizabeth and Edmund were all the grandchildren of Elizabeth Chenney. Elizabeth was married twice. Her only child from her first husband was a daughter who married the Duke of Norfolk and was mother to Elizabeth and Edmund. Wit her second husband she had 7 kids. It was her daughter Anne from this marriage who was mother to Jane's mother Margery.
Katherine Parr was of the wealthy landed gentry. Her father was a knight and lord of the manor. Her mother Maude was an heiress, and was connected with Catherine of Aragon. After the death of her second husband Baron Latimer, Lady Latimer (Katherine) used her mother's connections to gain a place in the household of Lady Mary, where she caught the king's eye.