This would be unlikely, but not impossible. The biggest challenge is how to accelerate healing.
I envision this disease as being very species-specific, and it will need to have at least two different phases. Perhaps a parasite would be an ideal candidate, rather than a bacteria or virus.
First, let's talk about how healing works, and then we can talk about an evolutionary pathway for the disease.
Healing in humans is very complicated. The "simplest" healing is repair to a cut or scratch. But there is also inflammation repair, fighting diseases, replacing necrotic tissue, etc. Our disease agent would have to either co-opt this process, or perform its own form of healing. Since humans (and mammals in general) have been evolving and perfecting our healing process for many hundreds of thousands of years, I'm leaning towards a custom healing process.
For example, our disease agent may be able to reconfigure arbitrary cells into stem cells, and then use those cells to patch damage. Our disease agent may also come with a slew of antibacterial and antiviral compounds, which is releases into the body. Other healing methods are possible.
The problem that we have is we need a disease that has two distinct phases: a healing phase, and a hurting phase. Normal bacteria and viruses are generally not complicated enough to support this distinction. Parasites, on the other hand, are.
I envision the first phase of our disease agent as being a slow reproductive journey to a "critical mass" of disease agents. Then, after it reaches critical mass, it switches to an attack mode to aid distribution of the disease agent.
But how could something like that come about?
I envision an evolutionary pathway as follows:
First, we have a parasite that, for its own protection, naturally evolves antibiotic and antiviral compounds that it releases into its host. This is advantageous because, not only does it protect the parasite, it also benefits the host, which makes the parasites home that much more resilient.
However, this poses a problem for the parasite, as the healthy host tries to fight it off, and it is unable to spread as easily. The parasite evolves to include an "attack" phase, where it cuts off its antibiotics/antivirals and initiates an inflammation response that causes the body to attack itself. This allows a rampant breed cycle that causes the parasite to shoot up in numbers, and when other humans try to help our victim, are spread via touch or water contamination.
Over time, the parasite perfects the healing routine. This is an evolutionary advantage, because a healthy host gives the parasite mobility, and limits any competition for the host's resources when the parasite goes "critical".
There may be other ways to explain this type of a disease, but this seems to me like the path of least resistance.