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This is actually Connecticut's policy -- voters are asked for ID (the definition is much more generous than PA's), but if they don't produce one, they are allowed to vote anyway. We require them to sign an affidavit that they are the person they claim to be, that they reside at such-and-such address, etc before giving them a ballot. If I recall, New Hampshire does something substantially similar to this as well.
The only exception is that under the Help America Vote Act, first-time voters who registered by mail without providing an ID with their registration (including partial social security numbers) will receive provisional ballots. States have some latitude about what offices appear on these ballots (just federal offices, or local/state races as well?) and when they are to be counted.
In any case, asking for but not requiring ID works quite well in many places -- it serves to single out potential problems for additional attention.