We report all-electron, density-functional calculations with large Gaussian polarization basis set of the recently synthesized octahedral O B24N24 cage that is perfectly round by symmetry, and boron-nitride (BN) clusters that its existence might suggest. We consider whether it is energetically possible that the two halves of this round cage could cap the BN nanotubes, modeled by B28N28 and B32N32. The energetics show that BN nanotubes with such round caps are only slightly less favorable than the BN clusters containing six squares as the only defects in the otherwise perfect hexagonal lattice. A larger B96N96 O cage formed from B24N24 by adding sufficient hexagons to isolate all squares is not very favorable energetically. Published by Elsevier B.V.