FDR had reasons for overconfidence. The landslide victory over Alf Landon in November 1936 wasn't only a win for him, but also for the Democratic Party, which upped its already substantial majorities in both houses of Congress. So the president unveiled his plan for adding, at the president's discretion, a new judge for every federal judge over 70, ostensibly to help the old-timers with their workloads. The proposal would have applied to all levels of the federal judiciary, not just the Supreme Court, but of course most of the attention was focused on the high court.
Opposition to the plan was swift and deep among not only the public but Congress, a vast Democratic majority notwithstanding. Roosevelt's motives were unspoken but transparent: the court had struck down major parts of the New Deal, and he was striking back. The president's ninth fireside chat, which he gave in March to promote the proposal, did nothing to change that perception.
The bill died after a few months, but FDR was able to leave his mark on the Supreme Court in the conventional way. In fact, from 1937 to 1943 he was able to appoint the eight new justices and elevate ideological ally Harlan Fiske Stone to be chief justice.