The first minister added: "I am not standing here saying, and we have not said at any stage, that some patients are not waiting longer during these winter times than we would want them to wait".
It continues: "The NHS is severely and chronically underfunded".
It said it is now impossible to meet the standards set out in the NHS constitution alongside fully recovering performance targets, consistently maintaining high-quality patient care, investing in the NHS's capital requirements, and joining up services to deliver 21 century care.
Liberal Democrat leader Mr Rennie said: "The pressure faced by the NHS has been coming for years, it was largely predictable".
"However, we are of course sorry for any extended waiting times experienced by patients and their families during this exceptionally busy period".
"We'd like to thank the public for their patience during this hard time and to our staff who have been going above and beyond coping in extreme circumstances".
"Extra medical and nursing staff were employed in all key areas and extra bed space was opened at Blackpool Victoria Hospital".
Nearly 15% of England's A&E patients waited more than four hours to be seen last month - the joint-worst figure on record - as senior doctors warn of patients dying in corridors.
The letter, leaked to the Health Service Journal, signed by consultants in charge of emergency departments in 68 acute hospitals across England and Wales, acknowledged the best efforts of staff, trusts and clinical commissioning groups.
In an unprecedented move, doctors told May that 120 patients a day are being managed in some hospital corridors, with "some dying prematurely".
People are waiting too long in A&E departments because there are no beds for them on wards, and because many of those hospital beds are taken up by patients who are waiting for their social care arrangements.
The letter adds that sometimes the consultants see over 50 patients at a time who are waiting to be shifted to a bed in the emergency departments.
NHS bosses apologised to the family, citing "significant pressures this winter because of a high number of complex cases and respiratory illness".
Answering questions after a speech in south London on Thursday, Mrs May said flu was putting extra pressure on services.
NHS Scotland figures and the health secretary, Shona Robison, pointed to an increase in flu as part of the reason for the poor results.
"The NHS today has launched their national flu programme and I would encourage people to act on the advice that the NHS is giving and also encourage NHS staff who haven't had the flu vaccine yet to have that vaccine".
The only exceptions to this are cancer operations and what the NHS called "time-critical procedures" where the surgery has to go ahead to avoid the patient's condition deteriorating further.
Professor Derek Alderson, president of the Royal College of Surgeons, said at the time: "Cancelling planned surgery is not a sustainable way of dealing with emergency pressures".
"I would assure Mr Wilson and all of our patients that we are monitoring the situation very closely and working hard to minimise delays". That has gone to 107 per 100,000 - that is four times the level of flu in this week past year.