It could. Yes, companies have been going toward high deductible + HSA plans for years now. This shouldn't be surprising given that even in a good year the % increase in medical losses to a health plan will exceed the average company's % increase in revenue.
If a company doesn't want to fork over 5+% (assuming the best) more annually on health insurance, there are only a few options, such as skimpier benefits, skimpier networks, higher employee premium-share, self-insure + stop-loss, and/or enter into a co-employment arrangement. I didn't include changing carriers, because that will only work a couple times and can be more trouble than it's worth.