How many times can a party postpone a deposition?
What recourse does a party have to prevent the first party from continually postponing?
Unfortunately this is done every day. "How many times" is a difficult question to answer. In some states there are discovery orders and deadlines that the court has set. If this is a Federal Case then the Federal System provides for discovery deadlines and the Federal Judges get very angry when deadlines are not met. There are severe sanctions.
If the matter has been dragging on then I would suggest that your attorney make a motion to compel the deposition if your state procedure allows. Get the issue before the Judge and let him or her read the other party the riot act and set a date certain OR ELSE. Good luck.