Michael, Tony is correct. MapInfo doesn't bother its browser window with all the details, so it clips the values for viewing purposes only. Nothing is lost from FLOAT, but on export the text file contains nothing more than what is seen in the browser window.
This is a sporadic issue for me, too, because I commonly need to gather info from a map and then take it to the field with me, hard-copy style. Example: perhaps I need to find a benchmark that I set in a terrain undergoing rapid landscape changes. If I can get within ten meters, using handheld GPS, I can always sniff the benchmark or recognize the detritus from the installation.
If you want to preserve decimal places, then TABLE-->Table Structure-->Maintenance, and within that pulldown you can reformat the columns. (I'm not in M.I. right now, so the actual pulldown sequence may differ from my memory.) In Tony's example, the 12 is column width, which accounts for the decimal places but also for the decimal point, the integral values for lat-long, and the minus sign (depending on where you're working, geographically). Thus (12,6) is a handy coding. If your data is gathered originally by handheld GPS, then you needn't show precision greater than 3 decimal places, although 4 always makes a person feel better. Most of us try to preserve the precision of original data, regardless.
Anecdotally, when printing for field visits, I modify tables to create columns for metric coordinates. My boots have a better grasp of meters than decimal degrees for distance to travel.
Dave Sherrod