NHS Shropshire Telford & Wrekin said that the situation was a result of "continued and unprecedented pressure on its services".
The incident was declared on Friday, and is the second time this year health bosses have moved to declare a 'system-wide' critical incident.
There have also been a series of critical incidents declared at Shrewsbury & Telford Hospital NHS Trust (SaTH) throughout the year – underlining the fragile nature of services provided at both Royal Shrewsbury Hospital and Princess Royal Hospital in Telford.
The latest situation means all areas of the county's NHS are under such levels of pressure that non-urgent cases are cancelled, to prioritise emergency treatment.
A spokesman for NHS Shropshire, Telford & Wrekin, said hospitals were facing "significant levels of Covid-19", and extended waiting times to access beds.
He added that people whose appointments are cancelled will be contacted – with all others to attend as usual.
He said: "We are continuing to see significant levels of Covid-19 in our hospitals, alongside high numbers of patients arriving at our hospitals for other conditions.
"There are extended waiting times to access beds within our hospitals.
"This, paired with difficulties in discharging patients due to a lack of capacity across our care sector as well as staff absence due to Covid-19 is causing a significant strain on the system.
"As a result, the healthcare system is taking additional steps to prioritise and maintain safe services for patients.
"Our aim is to prioritise patients with the highest level of need and ensure that we continue to be able to manage emergency care.
"This means that some non-urgent operations, where patients require a stay in hospital, will be postponed to prioritise patients with the most urgent clinical need.
"We regret that it has been necessary to take this step, but it is important that we focus on patients needing urgent and emergency care as a priority.
"If you are not contacted directly about an operation being postponed, please continue to attend your appointment as usual.
This critical incident is an indication of the serious pressure the system is facing. We are working extremely hard to ensure people are kept safe but there are ways that you can help.
"Our teams are continuing to work exceptionally hard, and we would like to reassure the public that despite the challenges faced, our services remain open for anyone who needs them."
SaTH, which has been in special measures since 2018, has been under increasing pressure in recent months, with issues at the hospitals – and with a lack of space in community care – leading to huge handover waits for ambulance patients, leading to major problems for the ambulance service in responding to emergencies.