For what it’s worth, I think House Republican staffers that Andrew cites below are right about the Obama veto threat.
More interesting than that, though, is the fact that the Statement of Administration Policy that includes the threat contains no substantive critique of the Boehner bill. Generally, statements like this (particularly when they contain veto threats) include several paragraphs (sometimes several pages) of argument about what’s wrong with the bill. The administration’s SAP about the Cut, Cap, and Balance Act earlier this month, for instance, listed numerous concerns about the bill, and ended with a presidential veto threat. Offering no arguments about the Boehner bill suggests the White House wants to leave open the possibility of signing this bill or one much like it, by avoiding committing to critiques of certain parts of it that would then make it difficult to sign legislation that included those parts.
In any case, I think there is zero chance that the president would veto a debt-ceiling increase that passed both houses of Congress. The politics of that would be horrendous for him, and there is no way he would do it. It’s an empty threat.