Thomas Robinson was the eldest of the nine children of William Robinson, a Yorkshire landowner. His brothers included Sir William Robinson and Rev Richard Robinson, who became Primate of Ireland. He married Elizabeth Howard, eldest daughter of the 3rd Earl of Carlisle, who died without issue. The family had acquired the Rokeby estate in north Yorkshire in the seventeenth century, and Robinson, who was a talented amateur architect, designed and built a new house in the Palladian style. He was financially extravagant, and in 1742 was sent abroad as Governor of Barbados, but was recalled five years later on the petition of the planter population. He had re-married there but his wife refused to follow him to London. He became director of entertainments at Ranelagh Gardens, but his overspending led to his having to sell Rokeby Park in 1769. On his death, his baronetcy passed to three of his brothers successively, and then by special remainder to Matthew Robinson.