I see why Tressler and Elkhorn were robbed and why Thaddeus was framed, but I don't get why a French bank was robbed. The only hint I got from the movie was "they delivered the note" and that makes no sense to me. Was Shrike French? And have I not been paying enough attention at all?
Apparently Rhodes (Shrike's son) held the bank in Paris at least in part responsible for not paying out the insurance money after his father died.
To quote Wikipedia (emphasis mine):
At the Pont des Arts in France, Dray is met by Rhodes, who reveals himself to be Shrike's son. He masterminded the Horsemen plot as revenge on those involved: Bradley, for humiliating his father; the Crédit Républicain de Paris and Tressler's company, who refused to pay the insurance on his father's death; and the company that produced the substandard safe used in the trick that led to its failure.
And here is the dialog excerpt from the movie (transcribed myself). It takes place between Dylan Rhodes and Thadeus Bradley during the final reveal in the police cell:
Daryl: I finally got a chance to really look into Lionel Shrike. His comeback attempt. His accident.
The insurer who denied the family's claim: Tressler Insurance.
The bank that carried the note: Crédit Républicain de Paris.
I also found that explanation a bit odd, because I don't think the bank would be able to pay out the insurance money on their own either way. After all, Tressler's company was the insurance company and responsible for approving any payouts.
Maybe he just was not very rational about this and his thoughts of revenge blinded him.
Maybe he just needed an excuse to rob a bank in a stylish fashion.