Of course, Houdini always did escape. But he also went out of his way not to humiliate his challengers. No escape was ever presented as easy, even when it was. Houdini would frequently say he had never faced such a well-made packing case or a more expert rope tie. He left good will in his wake.
The below is a nice example of this. This ran in the The Wisconsin State Journal following Houdini's suspended straitjacket escape from the State Journal building in Madison on January 23, 1924.
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Of course, there were exceptions, such as William Hope Hodgson in Blackburn. And later in his career Houdini did do some on-stage boasting. A 1922 review in the Baltimore Evening News complained, "Houdini spends about 10 minutes telling of his greatness, then performs two tricks that he has been showing for years." But in this case he was boasting of past triumphs.
Okay, so maybe Hollywood got it half right.