Some passengers waited for up to four hours for bags
Flights from Heathrow's new £4.3bn Terminal 5 are departing with hand baggage only after luggage check-in was suspended due to a processing backlog.
British Airways, which has sole use of T5, announced check-in of all hold luggage was suspended until Friday.
The airline has already cancelled 34 flights because of baggage problems and passengers have had to wait up to four hours to reclaim their luggage.
BA said it "sincerely" apologised to all those customers affected.
Angry scenes have also been developing in the airport's airside arrivals area, with pushing and shoving in "chaotic" passport control queues, the BBC's Priya Patel reported.
Travellers emerging from the baggage hall have told her that no luggage is coming through on carousels from arriving flights, and some complain that they have been waiting two to three hours for their bags.
Announcing the suspension of baggage check-in, Gareth Kirkwood, BA's director of operations, said: "We sincerely apologise to those customers who have suffered disrupted journeys or baggage delays.
"We always knew the first day would represent a unique challenge because of the size and complexity of the move into Terminal 5. We are working extremely hard on solutions to these short term difficulties."
The airline is advising passengers to check in with hand baggage only if they wish to travel later, or to telephone BA if they want a refund or to rebook a flight.
BA has offered passengers on cancelled flights £100 towards the cost of overnight accommodation, but travellers have reported that local hotels are charging up to twice that figure.
German businessman, Hertz Merkel, flying to Dusseldorf, was one of those left waiting in the long queues at the terminal's check-in desks.
"This is real British organisation for you. I've only got hand baggage but I still have to stand in the same queue. It doesn't look like I'll be getting home today."
BA TERMINAL CHANGES
27 March: Domestic and most European flights move from Terminal 1 to T5
27 March: Long-haul flights from Terminal 1 move to T5
27 March: Miami service moves from Terminal 3 to T5
30 March: Algiers moves from Gatwick to T5
30 April: All Terminal 4 long-haul go to T5 (except Singapore, Bangkok and Sydney)
17 Sept: Barcelona, Madrid, Lisbon, Nice and Helsinki move from Terminal 1 to Terminal 3
Early 2009: Singapore, Bangkok and Sydney move from Terminal 4 to Terminal 3
Maria Matheou, 20, a student from Greece who is studying in Leicester, said it was "ridiculous" to be told she could not take her bags.
"We can hardly go on holiday without our luggage. We did the check-in online at 3pm today, we got our boarding passes, we stood in line and then right at the last minute we were told we cannot take our luggage with us."
Check-in delays also meant that Ray Duffy, who was trying to return home on a flight to Edinburgh, was turned away from security because he was too late for his flight.
But after returning to the departure hall, he discovered he could no longer fly with his luggage.
"I just hope I get home. It's incredible. I've stood in line for an hour, I get to the front of the line and they just send me to the back of another line," he said.
The suspension of luggage check-in was just the latest problem to hit passengers hoping to leave T5 on its opening day.
BA earlier announced "initial teething problems" with car parking provision, delays in staff security screening and staff familiarisation had resulted in a backlog of baggage.
This led the airline to cancel a number of flights in and out of the terminal, including services to Munich, Frankfurt, Paris and Brussels as well as Edinburgh, Glasgow and Aberdeen.
Environmental campaigners in red T-shirts staged a peaceful protest
A further technical fault also meant seven flights left T5 without luggage on board.
The BBC's transport correspondent Tom Symonds said the backlog was caused by problems with the airport's three-stage luggage processing system.
The first stage - the fast bag drop - was working as planned, he said. But the second stage, an underground conveyor system, had become clogged up.
This was being blamed on staff failing to remove luggage quickly enough at the final unloading stage.
In a statement, the Department for Transport said it expected BA and airport operator BAA "to work hard to resolve these issues and limit disruption to passengers".
Local Conservative MP David Wilshire, who serves on the Commons Transport Committee, said it was not clear who was to blame - BA or the airport's owner, BAA.
"The problem in getting to the real truth of this.
"'Not my fault, guv, maybe his fault guv' is, I think, a bit of the problem," Mr Wilshire said.
"And we won't get to the bottom of this until we know a bit more. What we have to find out is why this has happened and make sure it's put right and hope that this is really a starting glitch. People have been let down, let's be honest about it."
Earlier, hundreds of protesters - wearing Stop Airport Expansion T-shirts and opposed to plans for a third runway and sixth terminal at Heathrow - carried out a peaceful protest at the terminal.
BA has set up an information line for T5 passengers: 0800 727800
Have you been through Terminal 5 today? What was it like? Do you work there? Tell us about the new terminal at Heathrow airport.
You can send pictures and video to: email@example.com or to send via MMS please dial +44 (0)7725 100 100.
Do not endanger yourself or others, take any unnecessary risks or infringe any laws.
The BBC may edit your comments and not all emails will be published. Your comments may be published on any BBC media worldwide.