Today, do we have miracles from God? Does God heal His believers if He so chooses, if it is His will or do we have what some call "through the natural order of things people are healed?" I realize we have no public healing today, but is the miraculous from God still in tact? Like someone who had full blown cancer, went to the doctor, was diagnosed with cancer, started taking the treatments, noticed something was wrong, went back to the doctor, and all cancer is gone. By the way this person was a believer in Christ, functioning, as it were, praying daily and doing all the necessary things we do. Was it from God or "through the natural order of things?"
The situation you offer as proof is not of the same order as the ones described in the Bible. People who were healed in the Bible had afflictions that all around them could testify existed, whether they believed or not. People were cured of high fevers, withered arms, blindness, deafness, leprosy, paralysis, and even death. Those cured had instant recoveries which were verified by all around them, whether they believed or not. And an important point that is often missed is that it didn't require belief on the part of those cured to be cured. Oh often belief was asked, but some were cured without even knowing who they were talking to. Some didn't even ask to be cured. (And some looking for a cure, even though they were believers, weren't given one.)
The problem is that the term "miracle" has been broaden to apply to any unusual event, or even normal events. The birth of a baby is called a "miracle of life." It is called a miracle for a person to survive a life threatening accident or illness.
Another problem overlooked is that in the Bible, incidence of miracles were never done simply for the miracles' sake. The miracles served as proof that the person involved was speaking on behalf of God (Mark 16:20; John 20:30-31). Yet, the word of God has been completely delivered and God said there would be no more (Jude 3).
Yes, God answers prayers (James 5:16-18). I've personally seen many things happen, seemingly by coincidence, in response to prayers. But I wouldn't label them as miracles. To do so would diminish the true miracles recorded in the Bible because what was recorded there certainly can't be called a mere coincident.
In your case you didn't address any of the following:
1) Did the doctor make a mistake in his diagnosis?
2) Did the patient respond more quickly than usual to the treatment?
3) Was it apparent to even non-believers that this person had cancer?
4) Was it apparent to even non-believers that this person no longer had cancer?
5) Was the cure instantaneous?
6) Was it notable who brought about the cure?
7) Was the cure complete, that is that there was no recovery necessary?
8) Did the miracle serve a purpose?
Your example did not rule out a natural recovery. A miracle overrode the natural laws of the universe in a notable way. What you described is where God's providence might have been involved, but you did not describe a miracle.