While I think there are probably some miracles recorded in scripture that fit that description, I think the overall picture we get is that miracles are events that demonstrate God's intimate connection with and lordship over the natural processes of this world. In other words, miracles aren't necessarily always supernatural events without natural explanation; sometimes they are natural events that occur at the direction and command of God. In other words, the miracle is often in the timing and effect of the event - in what it produces - than in the causation of it.
For instance, suppose there is a natural explanation for the parting of the Red Sea (I think there probably is, whether earthquake or tsunami or whatever.) It's still miraculous that this natural event occurred exactly when Israel needed it to. Even the Bible doesn't attribute the parting to supernatural causes. It speaks of a "mighty wind" that pushed back the waters. Wind is a natural cause. It's scientifically measurable. That doesn't make it any less of a miracle. What the event does, rather, is demonstrate that God is the Lord of nature. God created this whole world, set it all in motion, including all the natural processes that lead to things like mighty winds and earthquakes and the like.
Or take any of the Ten Plagues for example. They are almost all natural phenomenon - disease, locusts, frogs, etc. - and that's kind of the point. The point is to show that Yahweh, not the Egyptian gods, is Lord over nature. The miracle, again, is in the timing - that they all happened when they were most needed in order to persuade Pharaoh to let the Israelites go.
Or take something more recent. My atheist friend Hemant comments on this story on his blog, about a morbidly obese pastor who gets the motivation to lose weight after his thyroid imbalance suddenly clears up. He calls it a "miracle" and takes it as a sign to finally get disciplined about losing weight. My atheist friends, of course, point to all the natural causes (or medical misdiagnoses) that could possibly explain his sudden "healing" and renewed sense of discipline. However, on this alternative (I would say, more biblical) way of understanding miracles, a natural explanation for his healing does not exclude the possibility of a miracle. Why couldn't God have worked through whatever natural process caused his hormones to suddenly come back into balance? The miracle is in the results of the healing, in the effect it produced in that pastor's life. God is into changing lives, not doing magic tricks, after all.
I think the whole supernatural vs. natural approach to miracles probably springs more from the Greek, neo-Platonic dichotomy of the spiritual and the material that has crept into so many areas of Christian theology over the centuries. And as with so many other areas of theology (in my experiences) things start to become a lot clearer once you drop these dichotomies and return to a more Hebraic, biblical worldview that sees God and the natural world as inextricably interconnected - when you view God as the God of nature, not the God apart from or above nature. In a Hebraic worldview there is not "what nature does" and then "what God does", nature is what God does. As the old hymn goes "This is my Father's world."
Update: I continue my discussion of this topic in a follow-up post.