Was the original NYT article's claim that there was a 31.5% fraud rate concerning medicare in 2006 accurate?
tl;dr: No. The NYT article is about claims for "Durable Medical Equipment" that accounted for only 1.7% of all Medicare expenditures in 2006. (i.e., 31.5% of 1.7% is much smaller than 31.5% of 100.0%).
Or, yes, if "concerning medicare" applies only to "Medicare claims submitted by sellers of durable medical equipment." (see also note on "fraud rate" below)
As a result, Medicare did not detect that more than one-third of
spending for wheelchairs, oxygen supplies and other medical equipment
in its 2006 fiscal year was improper, according to the report. Based
on data in other Medicare reports, that would be about $2.8 billion in
improper spending. - NYT
The 2006 Medicare expenditure for "Durable Medical Equipment" (DME) was $6.9 billion (1.7% of the $403.7 billion in total Medicare expenditures for 2006) - National Health Expenditures, Table 4. In other words, if the 31.5% in "improper spending" applied to Medicare as a whole then the "fraud" would be around $124.4 billion more than the $2.8 billion stated.
Brief note on what "improper-payment rate" means
The National Health Care Anti-Fraud Association, an organization of
about 100 private insurers and public agencies, estimates that some
$60 billion (about 3% of total annual health care spending) is lost to
fraud every year, but that figure is considered conservative. In 2008,
government-wide "improper payments" cost the U.S. Treasury $72
billion, or about 4% of total outlays for the related programs. Of
that amount, 50% took the form of reimbursements to providers, medical
suppliers, and other Medicare and Medicaid vendors. Medicaid had an
estimated improper-payment rate of 10.5%, or $18.6 billion, for the
federal share of Medicaid expenditures — the highest rate of any
federal program. - source
Note that 50% of the 10.5% "improper-payment rate" was under-payment.
And is the interpretation that the fraud was intentionally hidden from auditors backed up by any evidence?
tl;dr: Not that I could find.
The NYT article does not support Coulter's claim that "Medicare had aggressively hidden the fraud from outside auditors." Statements attributed to lawmakers or their staff are worthless without corroborating facts. (Protections provided by the Speech or Debate Clause allow lawmakers and "congressional staff" to say pretty much anything - factual or otherwise. Coulter's protections flow from U.S. Const. amend. I).
The company that was doing the auditing at the time, AdvanceMed, is still at it and I could not find any evidence they had been investigated (or even given a stern talking to).
Nor could I find any evidence that the episode resulted in even a "shake up" at the Centers for Medicare and Medicaid Services.
Note on "fraud rate"
As mentioned in the other answers and comments, the terms "improper payment rate" or "error rate" and "fraud" are not interchangeable.
From Articuno's answer,
It is important to note that the improper payment rate is not a
"fraud rate," but is a measurement of payments that did not meet
Medicare requirements. The CERT program cannot label a claim
fraudulent. - source
From Black's Law Dictionary, 9th ed.,
fraud, n. (14c) 1. A knowing misrepresentation of the truth or concealment of a material fact to induce another to act to his or
her detriment. ● Fraud is usu. a tort, but in some cases (esp. when
the conduct is willful) it may be a crime. — Also termed intentional
fraud. 2. A misrepresentation made recklessly without belief in
its truth to induce another person to act. 3. A tort arising from
a knowing misrepresentation, concealment of material fact, or reckless
misrepresentation made to induce another to act to his or her
detriment. 4. Unconscionable dealing; esp., in contract law, the
unfair use of the power arising out of the parties' relative positions
and resulting in an unconscionable bargain. — fraudulent,adj.