The fault has forced the cancellation of tonight's southbound services from the cities to London.
The overnight Lowlander service suffered a "technical issue with the coaches" after emergency brakes came on and it was terminated at Stafford in the West Midlands.
It is understood to relate to a wheel problem.
Passengers were told: "Our engineering team is working hard to understand what caused the train to apply its emergency braking system."
The 173 passengers aboard the train were expected to arrive some five hours late after being transferred to coaches or other trains.
The fault comes as the latest of a series of problems with the new £150 million fleet, which entered service six weeks ago.
Since then, two other services have not completed their journeys - a northbound London-Edinburgh train was terminated at Carstairs while a southbound Edinburgh-London train was started at Carstairs.
Passengers have also complained of toilets not working and no water in basins and the ensuite showers.
One Glasgow-bound passenger, April Masson, who was in a seated coach, tweeted: "The most uncomfortable I’ve ever felt on a train.
"Limping along at 10mph on a #caledoniansleeper from Euston to Glasgow which can only be described as having a flat tyre."
She told The Scotsman: "At around 2:30am, the train did an emergency stop just north of Birmingham.
"Two staff came running down the carriage to one of the control panels in the carriage next to mine.
"Eventually the train started up again and there was a thumping sound coming from the undercarriage.
"As the train sped up the noise became louder and faster. Then the train stopped again.
"Staff continued to congregate around the control panel/area.
"There was no heating in the train so everyone was freezing.
"The train started and stopped a few more times before taking off at about 15mph. We continued at this speed (with the banging under the wheels) for about 45 minutes till we reached Stafford, where we were all decanted into buses."
Fellow passenger Dr Rachel Smith, a senior lecturer in phonetics at the University of Glasgow, told The Scotsman: "The Sleeper stopped very abruptly somewhere south of Atherstone [east of Birmingham around 2:30am, was stationary for the best part of an hour and then travelled extremely slowly and noisily to Stafford, where it arrived around 5:50am.
"Carriage lighting went on and off a few times during this.
"Leaving aside the train fault, I thought seats in the seated carriage were fairly comfortable, well equipped with sockets and individual lighting, and price increase on the seats is small (relative to berths).
"I passed three of the large-style toilets that were out of order, eventually to find a small one that was possible to access, though the door didn’t fully close or lock."
The first Caledonian Sleeper tweet at 4.56am said: "Our apologies to guests but due to a technical issue with the coaches, this service will have to terminate at Stafford.
"Replacement road transport has been arranged to take guests from Stafford to Glasgow. Our team onboard can advise further."
The company then posted another update at 5.09am saying: "Guests travelling onwards to Edinburgh can use connecting rail services from Glasgow, using their sleeper tickets.
"All guests can apply for a 100% refund through Delay Repay scheme.
"Our sincere apologies to all guests for tonight's service failure."
Another tweet posted at 7.17am added: "Our sincere apologies to all guests on our northbound Lowlander service, following this morning's cancellation.
"Road transport is now underway from Stafford to Glasgow and Edinburgh. Alternative train services are also available from Crewe."
It is the latest in a number of problems with the new service, which debuted at the end of April - but arrived three hours late in London following signalling issues on the line.
At the time, Network Rail said there was also an issue with the train itself.
Caledonian Sleeper managing director Ryan Flaherty said: “The Lowland service from London to Glasgow/Edinburgh was terminated at Stafford early this morning due to technical issues.
"We are working to understand the cause of this problem and we apologise to all guests affected.
"Once the service was terminated, we arranged for road transport to take guests to Glasgow while alternative train services were also available from Crewe.
"All guests who travelled on the service last night will, of course, receive a full refund.
“As a result of the technical issues, we have taken the decision to cancel tonight’s Lowland service from Glasgow/Edinburgh to London.
"We will be contacting all guests scheduled to travel to inform them of the cancellation and help them with alternative arrangements.”