Curtis Barnett's date of birth and his early service is not recorded; but he was already a lieutenant of some standing when, in 1726, he was appointed to HMS Torbay, Sir Charles Wager's flagship in the Baltic cruise of that year, during which he seems to have served on the personal staff of the admiral,[2] in a capacity afterwards known as a flag-lieutenancy.

Barnett continued in Bideford as part of the Mediterranean Fleet for three years, returning home in August 1734; and in the following February commissioned the 60-gun HMS Nottingham, for service as a guard ship in the Downs.

[2] On 1 August 1737 he turned over to the 60-gun HMS Dragon, and continued in the English Channel for some time after the declaration of war with Spain, when, in October 1740, he was sent out to join Admiral Nicholas Haddock off Cadiz.

Barnett on this sent a boat on board the Borée, to explain to the French commodore, M. de Caylus, that what had happened was due to the captain of the Aquilon, who had behaved with great want of politeness.

When Haddock was compelled by ill-health to leave the fleet, the command devolved for a short time on Rear-Admiral Richard Lestock, between whom and Barnett a difference of opinion gave rise to a correspondence which, viewed by the light of after events, seems to have an almost prophetic significance.

A few months afterwards Dragon returned to England, and in March 1742–3 Barnett was appointed to HMS Prince Frederick for Channel service, and was with the fleet under Sir John Norris when the French made their failed attempt to invade and were scattered off Dungeness, on 24 February 1743–4.

The year 1745 was spent in a vague cruise in the Bay of Bengal, backwards and forwards from Ceylon to the mouths of the Ganges; and though two 50-gun ships, HMS Harwich and Winchester, came out as a reinforcement, Deptford and one of the frigates were sent home with a convoy.

For the time being the war was at a standstill; and a few weeks before a French squadron appeared on the station, Barnett died onboard Harwich at Fort St. David, Cuddalore, on 2 May 1746, after a few days' sickness.