A systematic study of increases of flux of 10-, 30-, and 60-MeV protons shows that some are associated with flares, but many are more closely related to sudden commencement (sc) of a geomagnetic (sc) of a geomagnetic disturbance or to crossing of a sector boundary (sb). Some of these sc- and sb-related enhancements occur as second maxima of a flare-associated increase; and these have energy spectra closely similar to the spectrum of the primary increase, a finding indicating that these protons, too, are accelerated in the flare. Isolated sc- and sb-related increases can be regarded as extreme examples of second maxima, in which the primary maximum is below the threshold of detection, and we see only the maximum of the increase, which occurs 2 or 3 days after the initiating flare. As a group these small, slow increases of short duration tend to have a softer spectrum and a more symmetrical profile than increases that follow promptly after the flare, and their associated flares are distributed more symmetrically over the solar disk. The occurrence rate of these slow increases mimics that of major flares and of flare-related increases, and so we conclude that these protons are also flare accelerated. The failure of 10- to 60-MeV protons to behave in conformity with a general model of flare-accelerated protons moving in the interplanetary magnetic field appears to be due to the complexity and inhomogeneity of the field rather than to lack of sophistication of the model describing the interaction of field with protons. Adequate prediction requires consideration of magnetic configurations such as those characterized by sudden commencements and sector boundaries.