Some operations have been cancelled and extra staff drafted in to deal with a surge of patients over the weekend.
The Bristol Royal Infirmary, Southmead Hospital and Weston General hospitals are affected.
More beds are being opened but people are urged not to go to A&E unless they have a "life-threatening" condition.
Instead they are urged to go to minor injury units, chemists and GP surgeries, or to call the NHS on 111.
A black alert, the highest level, means hospital services are unable to cope with demand. The three hospitals were last on black alert earlier this month.
The number of people in hospital for more than two weeks is much higher than usual, said an NHS spokesman.
A surge of patients during last week's cold snap has added to pressure.
The black alert is likely to stay in place for days. Extra ambulance staff are being used to transfer patients into hospital, to prevent ambulances queuing and district nurses are being deployed to hospitals, to help find community beds for patients well enough to leave.
Bristol GP Peter Goyder said an increase in hospital admissions together with "a higher number of very ill patients who need to stay in hospital for longer" had contributed to the situation.
Analysis of the most recent figures suggests hospitals in England are struggling to meet the four-hour target for treating or admitting 95% of patients.
Three hospitals in Cornwall also went on black alert this month, as did two in Somerset.

The cockpit voice recorder from the accident off Shetland in 2013, in which four people died, was recovered by the Air Accidents Investigation Branch.
A judge ruled it was in the public interest and the interests of justice to make it available.
The British Airline Pilots' Association (Balpa) said it had lodged an appeal.
The  union's general secretary, Jim McAuslan, said: "The 2013 Supa Puma accident was tragic, and it is vital the AAIB gets to the root cause and has access to whatever data it needs.
"However, providing the data to the prosecutor and the police in parallel to the AAIB's investigation cuts across everything pilots and the broader flight safety community stand for.
"We cannot stand by while the court allows that to happen without pursuing other legal avenues to highlight our concerns and question whether it is the correct approach."
Prosecutors have been trying to establish whether anybody could be held criminally responsible for the crash.
The Air Accidents Investigation Branch (AAIB) has the right to initial access to the cockpit voice recorder but it routinely chooses not to hand over the material to other bodies - in this case the Crown prosecutors.
The BBC Scotland news website revealed last year an order was being sought by Lord Advocate Frank Mulholland QC at the Court of Session to access the data.
It was a rare legal move.
Lord Jones said he was satisfied that disclosing the data would have no adverse impact on current or future crash investigations - but that it could only be disclosed to the Crown Office and Police Scotland.
A total of 18 people were on board when the Super Puma crashed on its approach to Sumburgh.
Helicopter passengers Sarah Darnley from Elgin, Gary McCrossan, from Inverness, Duncan Munro, from Bishop Auckland, and George Allison, from Winchester, lost their lives.

Hazrat Mirza Masroor Ahmad, the Caliph of the Ahmadiyya Muslim Community (AMC), said his "heartfelt sympathies" were with the victims.
He was speaking at a Peace Symposium held at the Baitul Futuh Mosque in Morden, south London on Saturday.
Five people died and 50 were hurt outside Parliament, last Wednesday.
In a message Prime Minister Theresa May praised the conference, which she called "an annual reminder of your community's commitment to promoting peace".
MPs Paul Scully and Siobhan McDonagh were among the hundreds who attended the event at the UK's largest mosque.
The caliph described the attack in central London a "barbaric atrocity".
"No matter what terrorists may claim, under no circumstances are indiscriminate acts, or killings ever justified," he said.
He added that Muslims and non-Muslims "must stand up against all forms of oppression, hatred and use all our capabilities to try and foster peace in the world."
Three people were killed in Wednesday's attack when Khalid Masood drove his car into pedestrians on Westminster Bridge, before stabbing to death unarmed PC Keith Palmer.
Masood was then shot dead by police.
Hospitals with A&E departments around Bristol are back on black alert for the second time in a month amid "severe pressure" on services.
[NEXT_CONCEPT]
A ruling that investigators should hand over the black box from a North Sea helicopter crash to Scotland's top law officer is being challenged.
[NEXT_CONCEPT]
The world leader of millions of Muslims has condemned the Westminster terror attack calling it an "affront to the teachings of Islam".