Many beyond the Standard Model physics theories, including all supersymmetry (SUSY) theories assume the existence of two Higgs doublets. In these theories there are two electrically neutral scalar Higgs bosons, one heavier and one lighter.* By convention, the symbol H is used for the heavier one and the symbol h is used for the lighter one.
One of these could be a Standard Model-like Higgs boson such as the one experimentally found to exist at 125 GeV of mass. (Experiments have also increasingly ruled out the possibility that the 125 GeV boson is a mix of scalar boson with a significant mixing of a pseudoscalar boson to a very high confidence level.) But, there is little theoretical guidance on the question of whether the other would be lighter or heavier.
A recent review of Fermilab data has ruled out the existence of a light second Higgs boson at masses from 10 GeV to 100 GeV, which is an important part of the parameter space for many such two Higgs doublet models and strongly favors two Higgs doublet models in which the 125 GeV Higgs boson is the light, rather than the heavy of the neutral scalar Higgs bosons in the pair.
Previous bounds on neutral Higgs boson masses in two Higgs doublet models can be found here. This study significant expands the exclusion range in addition to making previous bounds more robust by replicating them with methods that are not identical. Experimental limits on extra scalar and pseudoscalar Higgs bosons are essentially the same because their decay channels show up in the same kinds of experiments.
* The other three additional Higgs bosons predicted in such theories are a Higgs boson with positive electric charge (H+), a Higgs boson with negative electric charge (H-), and an electrically neutral pseudo-scalar Higg boson called A. Limits on charged Higgs boson masses can be found here with new data from the LHC creating more strict exclusions found, for example, in this paper.
More elaborate more than two Higgs doublet models also have, for example, doubly charged Higgs bosons which are excluded up to much higher masses than singly charged Higgs bosons (they have masses of not less than 322 GeV if they exist).
There is no evidence for the discovery of any of these either and there are significant recent exclusions on their existences, although it surprises me that the possibility that the 750 GeV bump is a heavy neutral scalar or pseudo-scalar Higgs boson that produces diphoton decays via a triangle diagram involving the two charged Higgs bosons, hasn't received more attention yet. Perhaps other data of which I am not aware rules out this possibility which is quite elegant compared to many of the other proposals out there these days to explain it.
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