It didn't happen then, but that doesn't mean it can't happen now. For several decades, the Republicans have played political hardball to pack the courts in their own way [1]; I doubt that was a factor nearly a century ago.
[1] bitter obstructionism to maintain vacancies until they have the power to fill them with their own picks, selected primarily for ideological reliability.
Eh, if the definition of "court packing" is so narrow that it only covers things nearly exactly like FDR's proposal, then I don't consider it a very useful term.
If it makes you more comfortable, feel free to replace "court packing" in my comment with a term that's general enough to encompass FDR's proposal and the Republicans' recent tactics.