  Recently it has been suggested that top quarks, or very massive fourth
generation quarks, might surround themselves with a Higgs "bag" of deformation
of the Higgs expectation value from its vacuum magnitude. In this paper we
address the question of whether such nonlinear Higgs-top interaction effects
are subject to experimental test. We first note that if top quarks were
necessarily accompanied by Higgs "bags", then top quark weak decay would
involve the sudden disruption of the Higgs "bag", with copious production of
physical Higgs particles accompanying the decay. We then examine the effects
that such Higgs "bags" would produce on the spectrum of toponium, where the two
bound top (anti)quarks, and their "bags", overlap. We numerically evaluate the
effects that the nonlinear feedback in the Higgs-toponium system would have on
the energy level splittings of the toponium bound states, and find that for
allowed values of the top and Higgs mass the effect is negligible, thus
indicating that even in this favourable circumstance Higgs "bag" formation
around top quarks does not observably occur. Finally, we consider the case of a
second Higgs doublet, allowing the possibility of enhanced couplings for one of
the physical Higgs to top. Even in this nonstandard scenario the effects are
minimal, and we infer the general absence of observable effects at any level
that might suggest the utility of considering top quarks to be accompanied by
Higgs "bags".
