Are non-military aircraft allowed to fly with no registration numbers (tail numbers)? There have been several news reports about aircraft (light planes and helicopters) flying without visible registration numbers or identifying marks in the USA:


  The NYPD referred to the helicopter only as “23”—a reference to the
  number of police officers killed in the 9/11 terrorist attacks—and
  initially the aircraft had no registration number painted on its tail
  or side. source


and


  Cimbolic was beginning to worry that he had overreacted when he
  noticed, on the same flight-radar Web site, the second plane flying
  higher in the sky, carving bigger loops above West Baltimore. The Web
  site reported that this plane was a Cessna 560 Citation V, a small
  jet. But it showed no tail number, offering no possible trail to
  Federal Aviation Administration records. This only heightened his
  curiosity. source


Eventually these aircraft were identified (and in the case of the former, a registration number found). But the general question I have is:

Are non-military aircraft allowed to fly without visible registration numbers (tail numbers) in the United States?

Fine print: I am aware that military planes usually have visible tail numbers, this question is about non-military craft. So "non-military" in this case includes federal agencies and state and local police.
14 CFR 45 subpart C governs the display of "nationality and registration marks".

14 CFR 45.21 says:


  Except as provided in §45.22, no person may operate a U.S.-registered aircraft unless that aircraft displays nationality and registration marks in accordance with the requirements of this section and §§45.23 through 45.33. 


Section 45.22 gives some exceptions for "exhibition and antique" aircraft, and for aircraft of unusual design that may need an alternative marking scheme.

I presume there is another regulation, or international agreement, requiring that foreign-registered aircraft must also display their registration marks, but I haven't found it yet.

As to which aircraft have to be US-registered in the first place, see 14 CFR 47.  It's a bit complicated.  

The US military is generally not bound by FAA regulations (though they may choose to follow some of them), see How much jurisdiction does the FAA have over military aircraft?.