The Washington Post had an interesting article on how Pfizer learned that its arthritis drug, Enbrel, may be useful in slowing the onset of Alzheimer's disease. According to the article, Pfizer chose not to pursue any further testing of the usefulness of Enbrel as an Alzheimer's treatment, nor did it share its evidence with anyone else.
The most obvious reason why it did not pursue further testing itself is that the main patent on Enbrel was about to expire. This meant that if Pfizer invested the tens of millions, or possibly even hundreds of millions, needed for the clinical testing and FDA approval, it would likely be selling Enbrel as a generic competing with other companies who only have to cover their manufacturing cost. This means that Pfizer would never be able to recover the cost of the clinical tests and going through the FDA approval process.
The most obvious reason Pfizer didn't share its evidence with the scientific community is that it is a drug company that exists to make profits. It is not in the habit of making research publicly available to advance public health.
It is interesting to think of what this story might look like in a system of publicly funded research, where a condition of getting the money is that all findings are made available as soon as practical. (This system is described here and in chapter five of Rigged [it's free].)
In this case, the Pfizer equivalent that is finding evidence that the arthritis drug it developed would quickly post this on the web. Since its funding does not depend on patents, it may opt to pursue further research itself. However, if it is not well-situated to actually carry through the research on Alzheimer's, other researchers could pick up the ball and do the work themselves.
In the event that it proved to be a useful treatment for Alzheimer's the Pfizer equivalent would use this success as an argument for renewing and/or expanding its contract for research. While it is obviously better to actually test and prove the effectiveness of a new drug, finding an important discovery that laid the basis for further testing is a valuable contribution that should be rewarded. In this case, it doesn't seem the patent system would provide such a reward.