Not necessarily. As soon as the water evaporates, any minerals present in the water will be deposited on the surface of the object in question, whether it was previously dissolved or undissolved.
My point was that a whole-house sediment filter is only capable of removing undissolved minerals. If you want to remove the dissolved minerals as well, you need a water softener or reverse osmosis system.
Thanks for all of the responses. As with most things in my life, the answer cannot be a well defined, simple one. I am contacting a professionaltoday to come and diagnose the issue.
UPDATE: Thanks to all of you who responded. We went out to the property yesterday and discovered that we had no water at all. So, the time came to call a professional who quickly diagnosed the problem. It seems as if the bladder in the pressure tank had deteriorated from sitting (the house sat vacant for about a year) and because of that we were getting the pulsation and eventually zero water preassure. $375 and a new pressure tank later the system is working perfectly. I figured a post with the resolution to the problem would be in order should someone read this thread in the future. Thanks again for all the advice.