Ultimately, the question whether miracles are possible or not comes down to the question of God's existence. If God is real then miracles certainly are possible. If God isn't real then miracles are not possible as they are supernatural events (and therefore require a deity). I believe there is an abundance of evidence supporting God's existence starting with the Cosmological, Teleological, and Moral Law arguments.
There are many people who claim to have experienced or witnessed miracles as well. Finally, throughout the Bible we see many miracles take place. Why is that important? Well, the Bible is well supported historically and as a result seems to be a reliable document.
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Many people are being cured from incurable diseases, i.e. HIV, which if you think about it, they are being cured from incurable disease:
"an extraordinary and welcome event that is not explicable by natural or scientific laws"
That is the definition of miracle, we can conclude that miracles are possible but not by us.
If this was in another section I would say what we perceive as miracles do occur. However as this is religion, it means the question revolves around the existence of a god who allows miracles to happen. This is utter rubbish as there is no proof for a gods existence. As such miracles attributed to such a deity cannot occur. However, what we perceive as miracles "currently unexplainable" do occur i.e. someone gets well from an illness that they should not get well from.
Miracles is an event that takes place which cannot be explained by anyone and breaks that laws of physics. But if someone is cured by an incurable disease, e.g. HIV, it doesn't mean its a miracle, it just means that we haven't found a way to make a cure for it yet.
Everything happens for a reason; nothing happens because of just magic or some supernatural cause. There is some form of science and math behind everything that happens. I'm probably a little biased, considering I'm atheist.
Is a homeless man winning the lottery a miracle? No, it's just probability. He had the same chance as Bill Gates winning it.