How do physicists know that mass of possible Higgs particle is limited between two values 90 GeV/c$^2$ and 145 GeV/c$^2$?
As noted by @Lubos-Motl, at the time that this was asked we had a pretty good estimate for the range of masses the Higgs might sit at. People were tossing around preliminary results from ATLAS and CMS that were far from discovery quality, but were indicative to say the least (though some would argue over that, still). If you check the Arxiv for dates around December 2011, you'll notice a leap in the number of papers predicting a Higgs in this range ;)
To give you something more of an answer; because we did not find any evidence for a Higgs decay in the Tevatron and LEP data we were able to exclude masses below about 114 GeV. Similarly, Tevatron excluded a band around 158-175 GeV. See below:
There is a range of Higgs masses that are "nicer" theoretically in their relationship with other parameters of the standard model and the prediction you heard was probably based on this.
As it stands, we've seen a "Higgs-like" particle which has properties which are yet undetermined. Unofficially, it looks like it's spin 0 and positive parity...i.e. it matches the most obvious properties of a Standard Model Higgs. In reality the statistics just aren't good enough yet and we'll have to wait to see if it truly is the "Higgs" that we talk about so much. What we have now for the mass of our "Higgs-like" particle is below:
Technically, the Higgs (or something like it!) could still be at higher masses...but talking about that would require a trip to the PDG to fish out those exclusions ^^...maybe an edit later.