  Regular magnetic monopoles in the non-Abelian Born-Infeld-Higgs theory are
known to exist in the region of the field strength parameter $\beta>\beta_{{\rm
cr}}$, bounded from below. Beyond this region, only pointlike (embedded
abelian) monopoles exist, and we show that the transition from the regular to
singular structure is reminiscent of gravitational collapse. Near the threshold
behavior is characterized by the rapidly increasing negative pressure, which
typically arises in the high density NBI matter. Another feature, shared both
the NBI and gravitating monopoles, is the existence of excited states, which
can be thought of as bound states of monopoles and sphalerons. These are
labeled by the number $N$ of nodes of the Yang-Mills function. Their masses are
greater than the mass of the ground state monopole, and they are expected to be
unstable. The sequence of masses $M_N$ rapidly converges to the mass of the
embedded Abelian solution with constant Higgs. The ratio of the sphaleron size
to that of the monopole grows with decreasing $\beta$, and, at the same time,
both fall down until the solutions cease to exist, again exhibiting collapse to
the pointlike monopole. The results are presented and compared both for the
ordinary and the symmetrized trace NBI actions.
