Yes, but I don't think you can call it a "bromance" per se. For a time, brothers may get along; even famous brothers:
Harry was said to have made a light-hearted best man's speech, laced with jokes at his brother's expense and even some of his more elderly relatives, after an evening dinner hosted by Prince Charles to celebrate Williams' marriage to Kate Middleton in front of 300 guests.
Harry is even reported to have joked about the relative heights of Kate and the Duke of Edinburgh, his grandfather being dwarfed by the new Duchess of Cambridge in her three-inch heels.
Prince William and the Prince of Wales had earlier used parts of their speeches at the Queen's afternoon champagne reception to mock each other, continuing the tradition of father-son speechmaking.
The Prince of Wales drew attention to his son's baldness and, in return, Prince William highlighted his father's waistline.
His father said he hoped Prince William would look after him in old age before joking that William would "push his wheelchair of a cliff".When they fall out, the bromance ends. It is then replaced with a mad grab for power. Avoid having any kind of a bromance with a father or grandfather figure as well, especially if you are in the middling part of a hereditary line of succession.