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”.
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.”
Additionally, much of the free space required by the update is made available again once the installation process has completed.
In a separate development, Apple has taken steps to reassure users that it takes privacy seriously, by vowing that it would not hand over data to government authorities.
“I want to be absolutely clear that we have never worked with any government agency from any country to create a backdoor in any of our products or services,” said Tim Cook.
“We have also never allowed access to our servers. And we never will.”
The firm also emphasised that protected data stored on devices running iOS8 cannot be handed over to law enforcement agencies, as Apple does not have the option of overriding a user’s own passcode.