Tube services from London to Heathrow airport have been disrupted after a problem with a train door.
London Underground said a full inspection was carried out to check there were not any wider problems.
The Rail, Maritime and Transport (RMT) union said it had repeatedly raised concerns about trains on the Piccadilly line.
London Underground operations director Pat Hansberry said: "We are very sorry for the delays to Piccadilly line services this weekend.
"Following an isolated problem with one train door, we carried out a full inspection to confirm there wasn't a wider problem, which resulted in some cancellations.
"We are now working hard to restore a full service on the line. We will be paying automatic refunds to our customers who have suffered delays to their journeys."
RMT general secretary Mick Cash said: "RMT has repeatedly raised questions about the safe operation of the Piccadilly line fleet and the door incidents have thrown the issue right back into the spotlight.
"RMT safety reps are investigating the current situation and our absolute priority as always is the safety of passengers and Tube staff alike.
"There can be no compromising on safety while these issues remain unresolved."
The RMT said passengers heading for Heathrow were left stranded at Acton Town because of the disruption. The union said urgent talks were being planned between its safety reps and LU.
The Transport for London website said there were severe delays on the Piccadilly Line with no service on some sections, blaming a temporary shortage of train operators.
A message said: "Your tickets will be accepted on the local buses and Great Northern and Heathrow Connect train services. Please avoid using the line."
Pat Hansberry, London Underground operations director, said: "We are very sorry for the delays to Piccadilly line services this weekend. Following an incident with one train door, we carried out checks on all of the fleet which shows that this was an isolated incident.
"We have met with the trade unions today to give them full assurance that there are no wider safety issues.
"Despite these categoric assurances, several drivers are refusing to work. We are doing all that we can to get things back to normal and do everything we can to provide automatic refunds for customers who are delayed."