The lawsuit, which is seeking class-action status, says Motorola promised to upgrade the operating system for the phone, which was sold with an Android 1.5 operating system.
Google has since upgraded that system three times, the lawsuit says.
The Chicago Tribune reported Saturday that the customer, Jack Haught, is seeking to recoup his expenses for the phone, which he says cannot handle such popular Web sites as Facebook or MySpace. It is also unable to manage services, such as gmail, the lawsuit says.
There have been hundreds of thousands of Cliq XT phones sold. Motorola, the newspaper said, declined to comment on the case.