Director-General of Health Dr Ashley Bloomfield said this is the current capacity.
An independent audit by Dr Ayesha Verrall said that tracing was too slow and constrained.
That 5000 maximum would increase as resources were added, Dr Bloomfield said.
The Government said it will put $55 million more into the system.
In early April, the national centre averaged 450 contacts traced per day over a four-day period, with a high of 700 on one of those days.
National Party's health spokesperson Michael Woodhouse said it was not believable to think that capacity had risen by 10 times.
"There are 220 people in the contact tracing centre, and just 100 in the public health units.
"They're doing a number of things, not just contact tracing.
"So it's a mathematical impossibility for 5000 successful contacts to take place."
Factored in too, was the number of calls that failed because no one picked up or the number was out of date, which the audit shows had been running at 40% of calls made.
"Now unless they're able to fix that, the Government won't be able to have the confidence that they can lift out of level 3."
The ministry told RNZ in a statement on Friday that two-thirds of the contact details of all New Zealanders in its databases were correct, but one third "may be out of date", and it was still trying to find technology to fix that.
At yesterday's briefing, however, Dr Bloomfield said well over 90% of people's contact details in the National Health Index were correct.
In the first 10 days or so of tracing since the national centre began on March 24, out of 13,000 calls made in total, only 4000 or so contacts were found. The centre has added a few dozen staff since then.
The audit said the fast establishment of the national centre was "impressive". However, it found the public health units were the weak link, under great strain, with too few staff and needing an urgent boost to tracing capacity of 3-to-4-fold.
The audit was damning, but Dr Bloomfield had denied to the Epidemic Response Committee as recently as a week ago that there was "even a problem with resource", Woodhouse said.
Committee minutes show the Director-General said "we're certainly coping with the current" case numbers, though new cases by this time were low. He told the committee they wanted to be able to ramp tracing up, and were "well on the way" to being able to trace the contacts of several hundred cases a day.
The committee was told by Australia's head of public health that it had about 7000 staff doing contact tracing; that is about five times more per capita than New Zealand.
Dr Bloomfield also told the committee: "One of the nice things that Dr Verrall points out is that, actually, really good contact tracing can be as effective as a vaccine."
Late last week, the epidemiologist on the ministry's advisory group, Michael Baker, told RNZ that he had never seen the data on public health unit capacity, and doubted it existed in a readily analysable form.
This was confirmed by Dr Bloomfield on Sunday, when he said the analysis of the public health unit data had been held up by creaky localised information systems, so analysis of it had had to wait till the last minute at the weekend, in order to provide it to the Cabinet yesterday as part of its alert level deliberations.
The audit said the ministry should "rapidly develop" a smartphone app to help with tracing.
"It needs to be available quickly," Dr Verrall said.
RNZ understands that Auckland company Rush Digital is building the app for the ministry.
The ministry has not confirmed this, or said how it gave out this contract, despite repeated RNZ questions.
RNZ also understands that a Bluetooth-based tracing app, like Singapore's, might be delayed for use here till late May.
