Dover Lifeboat crew were called out twice to the same boat within 24 hours on Sunday.
The first call, at 2.20pm, was when the yacht had become partially dismasted and was to the east of Dover.
It had been seen setting sail from the port earlier in the day.
The lifeboat arrived alongside the yacht to find it was trailing various ropes, and entangled in one of its own sails. But the two people aboard the yacht refused all offers of help, and Mr Richardson and his crew had to leave it and return to Dover.
By 9.20pm that night, the yacht had drifted some miles westward and the Coastguard tug Anglian Monarch was standing by her. The crew of the tug could not see any signs of life on board the yacht and there was concern the drifting craft would run aground.
So the lifeboat was launched for a second time. This time two lifeboat volunteers went aboard the yacht and found the two men were alive. With careful persuasion the yachtsmen accepted the offer of a tow to safety. A rope was passed and the craft taken under tow.
The twin-masted vessel is believed to be of foreign origin.
Dover Lifeboat Coxswain Stuart Richardson said: “This was quite a frustrating day. There was so much we could have done to help the yacht’s skipper on the first call out but if someone refuses help we can’t force them to accept it.”
Early on Tuesday the lifeboat was called out again to go to the aid of a team of Channel swimmers who had successfully completed a relay swim to France.
The team had made the crossing in just under 14 hours, and were on their way back to England on their escort boat when the engine overheated.
Dover Lifeboat was called out at 4am, located the boat eight miles off Dover, and brought it back to the marina.