The meaning of obnubilate becomes clearer when you know that its ancestors are the Latin terms ob- (meaning "in the way") and nubes ("cloud"). It's a high-flown sounding word, which may be why it often turns up in texts by and about politicians. This has been true for a long time. In fact, when the U.S. Constitution was up for ratification, 18th-century Pennsylvania statesman James Wilson used obnubilate to calm fears that the president would have too much power: "Our first executive magistrate is not obnubilated behind the mysterious obscurity of counsellors…. He is the dignified, but accountable magistrate of a free and great people."
Latin obnubilatus, past participle of obnubilare, from ob- in the way + nubilare to be cloudy, from nubilus cloudy, from nubes cloud — more at ob-, nuance