I mean, if you count Mitch McConnell twice refusing to entertain spending bills that previously passed in the Senate, then yes, both branches are holding the government hostage.
And given that Mitch McConnell only has that ability by consent of the Senate. They could elect someone else in his place if they wanted to (Democrats might very well vote for a different Republican in such a case instead of voting party lines. it would only take a few Republicans siding with Democrats to replace Mitch).