Many have resorted to deleting photos, videos and other files in order to free up space for the new version of Apple’s mobile operating system, iOS8, which requires up to 5.8GB of storage.
Apple has also removed apps for its new health software because of a bug.
One expert said Apple’s updates were often prone to “teething problems”.
Some vexed Apple users took to Twitter to express their annoyance, at one point causing the subject to be trending above the Scottish referendum.
David Roberts tweeted: “This update would be great… If you didn’t have to delete half of the stuff on your phone just to install it.”
Daniel Zennon took a more humorous approach, tweeting: “So Apple put the #U2 album on everybody’s phone and then tell them they don’t have enough space for the #iOS8 upgrade”.
This is not the first time Apple users have had trouble with iOS updates.
In 2012, the iOS6 update caused some users to lose their apps, and others lost photos and messages when updating to iOS7 last year.
As well as requiring a lot of storage, the latest version, iOS8, does not include apps that run with Apple’s new HealthKit service, which is designed to work with third-party wearable health devices.
The software was originally scheduled for release in iOS8, but has been pulled while Apple works on fixing a bug.
David Price, online editor at Macworld UK, told the BBC the issues were not “really a surprise”.
“There’s always a rush on the servers on launch day, some delays, and usually some teething problems,” he said.
“That’s why we always recommend that people wait a day or two before updating.”
Apple users can avoid the need to free up storage space for the latest update by upgrading their software via iTunes on a Mac or PC, instead of through the phone or tablet itself.