High Deductible and Delays in Getting Care

I've been confronted about my very high deductible healthcare plan with a challenge that people with high deductibles delay care after diagnosis costing more for the insurer, be it Government or Insurance companies.

Here is the research:

And once diagnosed, women in high-deductible plans waited longer to start chemotherapy: an average of almost nine months among lower-income women, and nearly six months among higher-income women.

Diagnosis of breast cancer at early stages is associated with better clinical and survival outcomes. How the costs of care vary depending on the stage at which breast cancer was diagnosed has not been thoroughly examined.

Here is some research that goes the other way on screening and preventive care:

Sweeping statements about the cost-saving potential of prevention, however, are overreaching. Studies have concluded that preventing illness can in some cases save money but in other cases can add to health care costs

But the question is, can and will people learn if we were to implement a plan with very high deductibles? 

One data point that I go to is the Peltzman effect, how does the Peltzman effect work with safer cars? It may be by people hearing about someone being in an accident. Another is to consider the that the salience of terrorism has caused a huge overreaction in the USA. 
These lead me to believe that people can and will learn such things and that may tend move toward about or above the optimal reaction to a diagnosis healthcare regardless of the higher deductibles.

On a personal note I have paid a deductible of $10,000 in one case and of $30,000 in another and in neither case did we delay treatment. 

Also if the problem does persist, one way to combat delays might be to have health saving accounts. And there are many other potential ways to address the problem at low cost.