Yes, if you know someone and they can vouch for you and provide an introduction, that's a possibility, but you will need to have some prior experience that's relevant to the role.
The other way would be to be working at a management or strategy/IT consultancy that is working with the financial institution. That way you can build your own relationships, understand their business and get head-hunted into an internal role when one comes up.
Yeah, I figured that contracting is probably the only way I'll break in. I haven't ruled it out, though I really hate working for consultancies like that.
Maybe I should just get my mom to write them a note explaining how clever and handsome I am, because I don't know that that comes through clearly in the resume. If I attached that as a cover letter, it might at least be memorable :)