Hot-shot Mark West's 82nd-minute strike was enough to earn Thame a hard-won point on a heavy pitch at Nyewood Lane.
However, with closest rivals Bedford beating Northwood 6-1, Thame lead at the top of Ryman League Division 1 is now down to three points.
Thame started brightly on Saturday and took control early on with Nando Perna testing home keeper Will Packham.
But the home side gradually got into the game and forced the leaders on to the back foot.
Dave Birmingham broke clear of the Thame rearguard in the 23rd minute, only to see his effort well saved by keeper Ian Moores.
United lost influential midfielder Justin Merritt with a groin injury in the 35th minute, substitute Justin Lee fiulling in.
The visitors came under almost constant pressure, only some wasteful finishing by the Rocks strikers denying them the lead at half-time.
Thame reorganised and started the better of the sides after the break, but much as in the first half, they were soon forced to defend.
Bognor's pressure was rewarded in the 65th minute when Birmingham's cross was slotted home by Matt Russell.
The home side appeared to be heading for maximum points until West pounced with his 23rd goal of the season in the 82nd minute.
The experienced striker flicked on a long ball into the path of substitute John Mitchell, whose effort was blocked by Packham. The ball ran loose to West, who made no mistake hitting the target from ten yards.
Thame: Moores, G Williams, Potter, Brown, A Williams, Avery, Merritt (Lee 35), West, Swaysland (Mitchell 54), Perna, Minton (Fontaine 78).
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