The short answer is that you must either: have a large size version of your photo by uploading a large enough original or set your "who can download" permission in Flickr to "anyone". See below for details.

Here's what the On Black tool does when you choose to view Large:

1. If a Large version of the photo is available, it shows it. Photos only have a Large version if the original is bigger than 1280 pixels. Click the All Sizes button to see if a photo has a Large size.
2. If the Large size isn't available, it attempts to show the Original size. Accessing the Original size depends on the owner's permissions. In particular, the "who can download" permission. Note that if this is set to "Flickr users" then the viewer must have linked his BigHugeLabs.com account to Flickr.com to view the Original. If set to "anyone" then anyone can view it.
3. If both of those fail, the Medium size is shown.

What happens is that On Black saves the image location for a little while to enhance performance (so it doesn't have to keep asking Flickr where the image is).

When an image is replaced or the privacy is changed, the location changes, and it can take a just little while for On Black to automatically register the new version.

The most likely reasons is that you've opted out of BigHugeLabs.com. You can change your opt-out setting here.

Making your photos private will also prevent On Black from working properly. You will still be able to see them if your Flickr account is linked but no one else will.