Brighton and Sussex University Hospitals NHS Trust has told patient groups it is expecting to continue in that state all winter.
At this state of alert patient safety could be compromised, NHS England guidelines say.
The trust said its staff were "working hard" to give patients the emergency care they needed and give them a bed.
In a statement it said: "Our emergency departments are extremely busy with very high numbers of acutely unwell people.
"[This] is coupled with low numbers of patients who no longer need acute hospital care being discharged, and this inevitably impacts on the availability of beds."
The trust runs two acute hospitals, the Royal Sussex County Hospital in Brighton and the Princess Royal Hospital in Haywards Heath.  It treats more than three quarters of a million patients each year.
One patient, Barry Salvage, said he had to sleep on a trolley in A&E all night and it made him feel "terrible".
When he asked a nurse for bed, he was told they did not have any.
David Liley of Healthwatch said patients were being sent home "as soon as they are ready", increasing pressure on support systems in the community, which were forced to "swing into action".
Brighton and Sussex University Hospitals NHS is at the highest level of alert - known as Opel 4
NHS England's Operational Pressures Escalation Levels (Opel) Framework aims to bring consistency nationally in times of pressure.
It means the trust:
At least three other trusts in Kent and Sussex were at the highest level of alert last week - Medway Foundation Trust, Dartford and Gravesham NHS Trust and East Kent Hospitals University NHS Foundation Trust.
Patricia Marquis of the Royal College of Nursing said its members were "at the end of their tether".
She said: "They have been working at these really high levels of intensity for weeks and weeks."
Last year, Brighton and Sussex University Hospitals NHS Trust was placed in special measures after inspectors deemed it unsafe and poorly-led.
On Friday a spokesman said it was "making improvements" to help with the flow and discharge of patients.
He asked that people use A&E appropriately and considered "the many alternative services available".
A hospital trust has been on the highest possible state of alert every day since Christmas, it has emerged.