Mini Trip, Mini's are very sensitive to a failing battery, mine went with no warning and at first I thought it was a duff starter motor. On having the battery load tested and finding that the CCA was well below the batteries capability, a new battery was the correct solution. I don't however have IBS, that being said, one of my Mini's sat for over a month with previous short distance drives. Came home after work two days running and could hear a radio playing, turned out to be the Mini due to the battery running low on charge so as the ECU is also a parasitic drain on the battery, once they get low enough, they do weird and wonderful things except start the car. A 48 hr trickle charge brought the battery back up to spec and no issues with it over the last 12 months. Don't forget, your alternator is only to supply enough charge to the battery to maintain its level from the current demand and or starting the car. To fully charge and maintain a battery you need a minimum of a 24 hr trickle charge. Last of all, cold weather places an enormous load on a battery and can reduce its CCA capacity for extended cranking periods to start a car.
In regards of your question, yes to all in regards to a failing or extremely low charge state on your battery.
As a foot note, a lot of the new type of battery do not like a very low state of charge or to be completely discharged as this will damage and kill the battery so its prudent to maintain the battery charge state and keep an eye on the Magic Eye and make sure its in the Green, if its not, put it on a trickle charge asap.