According to this article:
When a list has more than 500 items, the filters automatically switch to show only "Show filter choices" option instead of a list of the values because of performance issues. After clicking the link, the script should add select boxes above the columns with the filter values.
In my client's case, after clicking on "Show filter choices" options, the system starts to obtain the data, but after some time it crashes with an unexpected error. I have consulted the sharepoint log and the error is following (it is a copy of the row from the log):
Process: w3wp.exe (0x06DC) TID: 0x1514 Area: SharePoint Foundation Category: Runtime EventID: tkau Level: Unexpected Message: System.Web.HttpException: Request timed out.