It depends on the circumstances. If the estimate was a firm estimate, where they committed to do the work for that price, then no--they have to honor the quote. But if the estimate was for information or by way of example or illustration only, or was simply a "good faith" estimate that was not stated to be binding--and particularly if they stated in the estimate or other documentation that the actual price will depend upon time spent on or complexity of the procedure--then they are not necessarily bound to the estimate. In that case, if they can justify the cost based on factors such as unexpected time or complications--they charge a higher amount.