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Found a reference concerning the original construction of the bomb shelter. Definitely was designed to protect against blast overpressure from a air burst in the D.C. area, but a direct detonation of a thermonuclear device? Even with subsequent improvements, I don't think so. The occupants would be vaporized.

This double-layer of stub girders shown in the White House reconstruction (1948-52) appears to fit the description of the top of a double-layered bomb shelter:
"We stopped all building in Washington when we built this bomb shelter. We took all the concrete and steel there was in Washington. These walls have three feet of concrete and steel; the top has three feet of concrete and steel, a space of three feet, then another three feet of concrete and steel." I went in it. This motor is supposed to filter this air for 48 hours to take care of 200 people. -- Ted J. Sanders

"Over the years, there have been several additions made to the main building, including the west wing (1902), the east wing (1942), and a penthouse and a bomb shelter (1952). The west wing, which contains the president's oval office and the offices of his staff, is the center of activity at the White House. During Harry Truman's presidency, from Dec. 1948 to March 1952, the interior of the White House was rebuilt, and the outer walls were strengthened. Nevertheless, the exterior stone walls are the same ones that were first put in place when the White House was constructed two centuries ago." -- Department of the Interior, U.S. National Park Service via Infoplease.