Apparently they can:

Consumer-driven health (CDH) products [i.e., high-deductible health plans relying on HSAs or Health Reimbursement Arrangements to reimburse for qualified expenses] have been marketed in various forms since the early 2000s. While emerging data is [sic] not entirely conclusive, general directional conclusions can be drawn from the studies published to date. […]

With regard to first-year cost savings, all studies showed a favorable effect on cost in the first year of a CDH plan. CDH plan trends ranged from -4 percent to -15 percent. Coupled with a control population on traditional plans that experienced trends of +8 percent to +9 percent, the total savings generated could be as much as 12 percent to 20 percent in the first year. All studies used some variation of normalization or control groups to account for selection bias.

For savings after the first year, at least two of the studies indicate trend rates lower than traditional PPO plans by approximately 3 percent to 5 percent. If these lower trends can be further validated, it will represent a substantial cost-reduction strategy for employers and employees.

Generally, all of the studies indicated that cost savings did not result from avoidance of appropriate care and that necessary care was received in equal or greater degrees relative to traditional plans. All of the studies reviewed reported a significant increase in preventive services for CDH participants. Three of the studies found that CDH plan participants received recommended care for chronic conditions at the same or higher level than traditional (non-CDH) plan participants. Two studies reported a higher incidence of physicians following evidence-based care protocols.

The authors add that “no data-based study has emerged” to contradict the indication that CDH plans “can produce significant (even substantial) savings without adversely affecting member health status”. H/T to Alex Tabarrok, who adds that the effects of such plans would likely be much more significant if they were adopted more widely.