Answer: I suspect you are losing the grafted tops and seeing suckers from the root stock. Once the part of the plant above the graft is dead, only the root stock will continue to grow. As a result, you no longer have a hybrid tea rose but a bush of whatever the root stock was, instead.
Many hybrid teas are not all that winter hardy. They may be suffering from a variety of problems that could weaken them and add to decreased chances of survival. Putting the cones on too early or too late, removing them too early or too late, and so on could also contribute to winter kill.
For more detailed information about caring for hybrid teas (including pruning and winter protection) you might want to take a look at a book or two on roses. One I particularly like is "Roses for Dummies" by Lance Walheim, Dummy Press, 0-7645-5031-4.
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