The resilience of the Antarctic Ice Sheet and its effect on global sea level depends on the dynamics of ice streams. Antarctic ice streams are known to be responsive to changes at the ocean interface and, as expected, most have thinned in response to ocean warming and sea-level rise since the Last Glacial Maximum (LGM). Here we provide direct and unexpected evidence that points toward the glacial/interglacial stability of the Slessor and Recovery glaciers, ice streams of the East Antarctic Ice Sheet (EAIS) which merge with the West Antarctic Ice Sheet (WAIS) to form the Filchner Ice Shelf in the Weddell Sea embayment. Cosmogenic-nuclide measurements in the Shackleton Range suggest that the Slessor and Recovery ice streams were not significantly thicker than today during the LGM. We hypothesise that the glaciers did not thicken because the grounding line was not able to migrate seaward beyond the deep Thiel/Crary Trough beneath the Filchner Ice Shelf immediately offshore. This discovery reveals how a topographic threshold can affect the dynamics of ice streams. It also reduces uncertainties on the thickness, extent and volume of the Antarctic Ice Sheet in a large but unknown sector of the Antarctic Ice Sheet; it constrains the potential sea-level rise from Antarctica; it helps explain observed anomalies in glacio-isostatic adjustment; above all it suggests that the behaviour of the Atlantic-facing Weddell Sea sector of the WAIS contrasts with that of the Pacific-facing Ross and Amundsen Sea sectors.