Popular problem on the '01's, as suggested earlier get them to check the fuel senders.

They eventually lemoned my "01 as it happened 9 times in some very sticky situations where the fuel guage got stuck at quarter to half a tank full and we ran out on Freeways/intersections and the last straw was 100 yards from home after driving across the country for 6 weeks

Best of luck.

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Currently 2010 X6 35i, Premium, Technology, Sport.
2004 Lotus Esprit - Final Edition
Former '04 X5 4.4i, Highland Green, Nav, DSP, PDC, Heated FR & R Seats, Comfort Seats, Rear Load Floor, Premium & Sport Package, IceLink Plus, Bel RX65, Tow Hitch, Sirius Satellite Radio.
Former '03 X5 3.0, Nav, Premium, Rear climate. (Stolen)
Former '01 X5 3.0, Nav, Premium, Rear Climate (Lemoned)