Not only was there such a period, but it lasted up until the LHC detection of the Higgs boson in 2012, and in a way it continues until today. Higgs boson is the quantum carrier of a scalar field, the Higgs field, and other elementary particles acquire mass through symmetry breaking by interacting with it, this is called the Higgs mechanism. The indirect experimental evidence for Higgs was quite compelling since W and Z bosons were detected at CERN in 1983 with masses as predicted by the theory, and especially after the electroweak precision tests that followed since 1989. But there were always theoretical issues with this particle (see below), and direct detection proved elusive for about 40 years.
Since 1979 this led to the development of a number of alternatives. Although known under the name of Higgsless theories many of them include an effective version of the Higgs boson, and therefore are not ruled out by the LHC detection. Moreover, some physicists almost rooted for Higgs not to be found. They saw finding "Higgs and nothing else" at LHC as a nightmare scenario, "because that would mean there wouldn't need to be any new physics all the way up to the Planck scale", according to CERN's Ellis.
Here is a typical sceptical account from 2011 pointing out that the predicted particle has not been found after surveying 95% of its suspected energy range. Of course, after the LHC detection the next year few doubt Higgs boson's existence anymore, but the Standard Model makes further claims about it, that are still in dispute. Namely, that Higgs boson is a fundamental particle, like photon or gluon, rather than an effective manifestation of some deeper physics. This is emotionally charged because in the latter case much of its hype, including "the God particle" nickname, would be misplaced.
The Higgs mechanism was not immediately accepted after being proposed either. The seminal 1964 papers by Brout-Englert, Higgs and Guralnik-Hagen-Kibble, anticipated by Anderson in 1962 in a non-relativistic setting, "were at first largely ignored, because it was widely believed that the (non-Abelian gauge) theories in question were a dead-end, and in particular that they could not be renormalised". "Essentially no-one paid any attention" also in 1967, when Weinberg and Salam independently showed how the Higgs mechanism can break the symmetry in Glashow's electroweak theory. According to Politzer, the mainstream acceptance came only after 1971, when 't Hooft and Veltman showed renormalisability of Yang-Mills gauge fields, and Lee developed and popularized their ideas.
In Higgsless theories symmetry breaking is effected by new interactions that manifest above the electroweak scale, rather than by an extra scalar field. The most popular ones appear to be technicolor, first proposed by Weinberg and Susskind in 1979, and extra-dimensional theories. Early technicolor was ruled out by the electroweak precision tests, but not the later "walking technicolor" versions, like 2004 one by Sannino and Tuominen, and extra dimensions, first advanced by Arkani-Hamed, Dimopoulos and Dvali in 1998, remain as popular as ever.
This recent paper gives a detailed analysis of theoretical objections to the Higgs mechanism of the Standard Model, that stimulated doubts and motivated development of alternative theories, from the modern perspective. The principal ones are.
Free parameters problem: Neither Higgs field parameters, nor its couplings to other fields can be derived from the Standard Model, they have to be fitted to the experimental data. That includes the mass of the Higgs boson, which is one reason why it took so long to detect it. Because "with four parameters I can fit an elephant", according to von Neumann's quip, many physicists find the theory distasteful. Similar objections are raised against the string theory.
Scalar field problem: This objection goes back at least to 1979, when it was raised by Slavnov. The Higgs field is the only scalar field in the Standard Model, the rest are gauge fields, and in the absence of Higgs they predict a number of zero mass particles, which are not observed. All other known scalar fields are only effective, not fundamental. So to some the Higgs field looks like a "convenient hypothesis to save the appearances" rather than a fundamental object. Moreover, scalar fields in general have serious conceptual problems.
Fine tuning/ hierarchy/ naturalness problem is perhaps the most severe one, and is a consequence of Higgs field being a scalar field. For scalar particles quantum corrections to the bare mass are particularly large, and 34(!) orders of magnitude of them will have to be canceled out by the bare mass to produce the measured physical mass of $125$ GeV. This miraculous fine tuning is similar to that in Lorentz's explanation of the Michelson-Morley experiment, where the lengths contract and the clocks retard by just the right amount to make the ether wind undetectable. Both technicolor and extra-dimensional theories were primarily aimed at avoiding this fine-tuning. By the way, supersymmetry was also proposed as a way to protect the small Higgs bare mass from huge quantum corrections.
All three issues underscore the idea that the Higgs mechanism has an air of ad hocness to it, and none of them is addressed by the LHC detection. It seems that the Standard Model is following Quantum Mechanics into becoming a victim of its own success. It describes everything we observe so well that there is no hope of experimental guidance to resolving its conceptual problems in the foreseeable future.