Did you use any gps Utility (like "GPS Test" or "InViu GPS-Details") to watch how many Satellite were in sight? (or used)
Did you test the GPS-Functionality outside under free sky or inside a building.
How long did you wait for the position to become available?
"Positioning while using internet" often utilized completely different techniques. Google/Android can calculate the position by known WiFi-HotSpots (which may have been unknowingly collected by other android devices passing by with gps&wifi enabled or streetview cars) or the cell-ids (which is much less exact).
The AGPS Function (internet support) usually ownly reduces the time until GPS is ready to provide coordinates (time to first fix). Usually also such AGPS Data is vaild for a week or so after downloaded once. (The App "GPS Test" has an option to manually request AGPS Data which makes only sense if internet is available.)
Getting a first fix for GPS, especially if the last successful use of GPS has been some time ago can take some time. So getting outside, enabling GPS, mantaining a steady position and waiting up to 15-30min may be required. The Apps "GPS Test" or "GPS-Details" may be used for watching the progress the receiver makes. Usually the satellites should show up nearly immediately but GPS then takes some additional minutes until ready. This is the time needed to download the almanac and ephemeris data from the satellites. (If not previously obtained by some AGPS function).