The Welsh government has urged people to think before dialling 999, as A&E departments across Wales continue to operate under "unprecedented" pressure.
Some hospitals are treating more patients than ever, and ambulances have been seen queuing with patients at A&E departments across the country.
The College of Emergency Medicine says staff levels are critical.
NHS and Welsh government officials are holding twice-daily conference calls to try and address the problem.
More than 450 patients were seen at the University Hospital of Wales, Cardiff, on Monday - a quarter more than expected - with almost half needing high-level care.
A new record was set at the A&E department at Morriston Hospital in Swansea on the same day according to the Welsh government - with more than 300 patients turning up.
Health Minister Lesley Griffiths said: "The public have an important part to play at this challenging time for emergency care services by using the most appropriate healthcare service to their needs and thinking before attending the accident and emergency department or dialling 999."
Patients are advised to use the Choose Well advice service to try and relieve the pressure on hospitals by contacting the best part of the health service for their needs.
The College of Emergency Medicine has told BBC Wales all emergency departments across the country are experiencing sustained and severe pressure, with a steady increase in seriously-ill elderly patients.
It says the situation has been compounded by the "worst access block" many of its members have experienced, with reduced bed numbers and staffing levels.
The college also says it is concerned mechanisms are not in place in the community to deal with demand.
The Welsh Blood Service said it was meeting demand from hospitals and had not seen "an increase in demand for extra blood products".
Chris Thomas, chair of the Welsh board of the Royal College of Nursing, said the usual "winter spike" in emergency cases seemed to be "bigger than ever before".
Allison Williams, chief executive of Cwm Taf Health Board, said there had been a rise in demand and a "large number of very ill patients are being brought to our accident and emergency departments".
She added: "We have experienced a sustained period of pressure since the start of the year and are working hard to ensure our patients receive timely and appropriate care."
The Abertawe Bro Morgannwg University Health Board said emergency departments across south Wales "are seeing an increase in the number of patients arriving who are seriously ill", and it had made 80 extra beds available.
Hywel Dda Health Board said some operations at its hospitals in mid and west Wales had been postponed as a result of the high number of hospital admissions, and it too has increased bed numbers to cope with the influx.
The Betsi Cadwaladr University Health Board said it had cancelled or postponed some operations due to "high demand" in recent months, and was reviewing the situation daily.
Welsh Liberal Democrats leader Kirsty Williams said the NHS in Wales was "teetering on the brink after over a decade of mismanagement by the Welsh Labour government".
Darren Millar AM, the Conservatives' spokesman on health, said the current situation was a "perfect storm for a cash-starved health service already hanging on by its fingernails".
Health academic Prof Marcus Longley, director of the Welsh Institute for Health and Social Care at the University of Glamorgan, told BBC Wales proposed closures of some A&E units could make health hot spots worse.
He said: "It could certainly increase some hot spots, there's no doubt about that, because if it's a 10 minute journey to the A&E department, in the future it will be 20 minutes.
"That's an ambulance that's tied up for that much longer if the call was unnecessary.
"I think all of that emphasises the importance of getting the key bits of the system right.
"If all you do is change bits here and there, that doesn't really solve the problem.
"But the health service is struggling to know what to do about this."