According to §91.175(c)
Except as provided §91.176 of this chapter, where a DA/DH or MDA is applicable, no pilot may operate an aircraft, except a military aircraft of the United States, below the authorized MDA or continue an approach below the authorized DA/DH unless—
So why are military aircraft exempt from this restriction?
The most simple reason is the military has its own set of flight rules (AFI 11-202, OPNAVINST 3710.7U) that for many years duplicated many of the applicable FAR/AIM regulations. In addition, the varied mission sets and training scenarios the military operates under sometimes require them to do non-standard things, so there are a number of "carve-outs" for them to perform and regulate their own maneuvers. In general, any exception to these rules are rare, unless specifically laid out in an instruction, and likely to be used only in times of national emergency or operational necessity.
One example of a reason to descend below an MDA or DA would be when shooting a self-contained approach. A self-contained approach (SCA) or IPRA (Independent Precision Radar Approach) is one where the aircraft uses its own navigational capability (INS/Radar/Radio Altimeter) to guide the aircraft to the runway without ground-based navigational aids. Referencing the old AFI 11-202v3 AFSOC Sup, it details when constructing SCAs,
7.4.6.5 For AFSOC SCAs and IPRAs, the MDA/DA will be commensurate with aircraft equipage capabilities and mission requirements. The applicable obstacle clearance will be maintained until descent on the final approach segment.
While practice requirements for SCAs require good weather, operational weather minimums will be much lower, and specified by the MDS-specific operations procedure. This may in some cases allow an aircraft to shoot an approach to a runway and descend below any MDA/DA on an existing approach, based on on-board capabilities.
