Tina Donnelly issued the warning after it was revealed Welsh health boards have spent more than £190m on agency medical staff over the last four years.
Tory Shadow Health Minister Darren Millar has written to the auditor general asking him to investigate.
The Welsh government said health boards should address agency spending.
The figures, obtained by the Welsh Conservatives through a Freedom of Information request, shows total health board expenditure on non-contracted hospital staff, including doctors and nurses, in Wales increased from £40m in 2011/12 to more than £71m in 2014/15.
Betsi Cadwaladr University Health Board spent the most each year with 2014/15 reaching almost £30m, double what it spent in 2011/12.
Over the same period the Powys Teaching Health Board, the smallest in Wales, saw its bill for agency staff increase from £170,000 to £635,000.
All of the health boards said they have found recruiting new doctors and nurses difficult.
Ms Donnelly said: "We have not been training sufficient nurses over the last number of years to meet the demands of the health care agenda so we do have to look at the workforce planning.
"Part of the agency expenditure is that we haven't seen a pay increase for nurses apart from 1% which is coming this year.
"Nurses are finding it really, really hard to cope so instead of going on a bank system within Wales they are now seeking to do their extra shifts with an agency because they pay more."
Mr Millar said: "Cash-strapped health boards are squandering tens of millions of pounds each year on agency fees because of a failure by Welsh Labour government to address chronic staff shortages in the Welsh NHS.
"Not only do hardworking doctors and nurses suffer when staffing levels are dangerously low, the quality of care can only be compromised by stretching resources to breaking point."
A Welsh government spokesperson said: "We expect health boards and NHS trusts to address agency and locum spend via a number of different ways, however we must recognise that agency staff play an important role in the Welsh NHS.
"The figures in this FOI represent a small percentage of the £6.7bn budget we spend on health and social services in Wales every year."
Not enough nurses are being trained to meet health care demands, the director of the Royal College of Nursing Wales has said.