People have been asked to stay away from A&E departments and hold off from calling an ambulance unless they have a life-threatening emergency.
The Welsh Ambulance Service said Christmas and New Year was its busiest time and asked people to know their limits while drinking alcohol on New Year’s Eve to avoid extra demand.
“We don’t have an endless supply of ambulances,” a spokeswoman said.
Health boards said their emergency departments are “very busy”.
Louise Platt from the Welsh Ambulance Service said: “The traditional party night puts more demand on 999 call handlers, ambulance dispatchers, clinicians in the control room and ambulance crews than any other time of the year.
“Whilst most people will drink responsibly, unfortunately there are some who will overdo it.
“From causing a scene in a bar, falling over and breaking your leg, to driving home over the limit, we urge everyone to stop and think.”
What is the picture around Wales?
Andrew Carruthers, director of operations at Hywel Dda health board, said it has had to manage two outbreaks of infectious illnesses at Carmarthenshire hospitals which affects its ability to discharge people.
“Our emergency departments are currently experiencing significant pressures,” he said.
“We have clear expectations about seeing and treating patients the way we want to, but we can’t do it alone and we really need the public to help us.
“The NHS is here for all of us and we are extremely lucky to have it, but if we want it to look after us we’ve got to make sure we look after it too.”
Aneurin Bevan health board said it had opened extra capacity to help manage an increase in demand.
“Our hospitals remain under a great deal of pressure,” a spokeswoman said.
“Our advice to local people remains the same – if you attend one of our emergency departments with an illness or injury which would be better treated elsewhere, you are likely to face longer waits because our staff need to prioritise the care of very sick patients.”
Swansea Bay health board told BBC Wales hospitals, especially the emergency department at Morriston, were “very busy” and warned of long waits.
“Clinical staff will always prioritise people who are very seriously unwell, so there may be long waits for those who do not need immediate attention,” a spokesman said.
Cwm Taf said staff were working exceptionally hard at its busy A&E departments, and asked people to support them by choosing the appropriate health care service for their needs.
Cardiff and Vale tweeted over the weekend that its emergency unit was “incredibly busy” and warned of longer waits there too, and Betsi Cadwaladr also said its emergency departments were busy over the winter period.
Health boards are encouraging people to use the Choose Well website to determine which NHS Wales service is right for them.
If it is not an emergency but still urgent, people can contact the NHS Direct Wales symptom checker on 111, as well as pharmacies and minor injuries units.
Figures published this month showed accident and emergency performance at hospitals in Wales was at a record low for the third month running and the Welsh Ambulance Service failed to meet its response time target for the first time in four years.
There were nearly 86,000 patients using A&E in Wales in November – 74.4% were seen within the four-hour target. For ambulances, 61.4% of life-threatening red calls arrived within eight minutes. The target is 65%.