Losing bottles and bottling out

According to this site (linked to by FF in a comment to Andrew's answer), the following are all possible origins for the term:

1. Cockney rhyming slang: bottle = bottle and glass = arse. To lose one's bottle = lose one's arse, i.e. bowel movement = show extreme fear = lose courage. Therefore, to have bottle is to have courage; to bottle out is to show cowardice.
2. bottle = bottle and glass = class = merit or distinction which, in Cockney terms, would include an ability to stand up for oneself.
3. Those who find these explanations over-elaborate prefer to locate the origin in the bottle-holder who acted as a second for a prize-fighter, using both the contents of the bottle and other skills to keep up his man's fighting spirit during a bout.
4. The simplest and probably the best explanation is that bottle originally stood for the courage that comes out of a bottle and has gradually come to mean genuine courage.

OED has an example of a negative use:

bottle, n.²

1.g. (d) Courage, spirit, ‘guts’; esp. in phr. to lose one's bottle, to lose one's nerve.

It has a note that "this use probably derives from the phrase no bottle ‘no good, useless’. It is however often popularly associated with the rhyming slang term bottle and glass = ‘arse’ and other similar expressions."

It definitely has positive uses, and there is a citation:

1969 It 4 July 11/2 You've gotta have a helluva lot of bottle to do something like that, and I believe that Morrison did it out of sheer contempt.

You might also remember the Milk Marketing Board's slogan "Milk's gotta lotta bottle".

I have always thought that it was related to courage via Dutch courage where one needs to be bolstered by the bravado which alcohol can bring; alcoholic drinks come in bottles.