Not so.

The offline capabilities depend upon the app, and the data the app is working on. Offline mode wouldn't work very well for something like a music player if your songs existed in the cloud, but would work fine if you had tracks cached in storage or on an SD card! For a game like Angrybirds*, it would be just fine -- assuming that the cachable data was small enough to be saved as well as the game itself.
* as it turns out, angrybirds will in fact work offline.

Where you draw the line, as you say, depends entirely on the app developer on a per-app basis.

For example, if I was coding up a text editor, I would cache recent documents (up to a point -- say 1MB), undo lists, user settings, and of course the application. This would allow people to continue to write text even if there was no internet connection. However, the major benefit is simply this: the next time the users hops online, they get the latest version of the app, and their data (if it's part of the application) is synced to the private server. They never need to update, install, maintain their app. This would come as a relief to casual users!

Other uses of offline apps include: vector map viewers, office suites, development IDEs, drawing/art programs, casual games, productivity apps (eg. todo lists), recipe viewers, ebook readers, etc. This would even be useful for pretty much any mobile app you can think of that doesn't rely on the internet. They typically are small and operate on small data sets.

To reiterate: offline capability must be specifically designed in the app itself and as such will be included if the developer sees value in offering it. The app cannot rely on the internet for functioning in this mode. It turns out, many applications are both small themselves and operate on a small data-set to garner offline support.

This is a feature of HTML5 and not chrome specifically. Firefox, Safari, Opera, and the upcoming IE all support HTML5 offline apps.