E.g. you don't want to refuse to open a document just because ten pages in there is a footnote that is somehow malformed.

Depending on the possible consequences, GIGO is much better for usability. Ideally you'd warn the user that something unexpected has occured and he should manually check the result but whenever you are handling external data where malformed input cannot be avoided then erroring early will only make your software less useful to the user who more often than not cannot do anything about the error but can manually correct the fallback if you got it wrong.