Many php applications, like WordPress Backup to Dropbox, require an extended max execution time and memory limit PHP settings in order to complete a task. By default the time setting is set to 30 seconds and, in the case of my WordPress plugin, is not enough for a backup to be completed.
If the PHP installation was not compiled in safe mode then a developer can temporarily change this value within a plugin using set_time_limit. So for any long running scripts setting the time limit to zero will tell PHP to allow the script to run forever.
However if your PHP installation is running in safe mode, this functionality is blocked and there is no way of altering the time or memory limit from within the application. Therefore the only way to get the an application to complete its task the php.ini max_execution_time and memory_limit setting will need to be altered. In the case of WordPress Backup to Dropbox this setting needs to be set to at least 3600 seconds (1 hour) and at least 150M respectively.
How easily this value can be changes depends on how your hosts PHP installation is configured. If your host allows it you can put a file named php.ini inside the root and wp-admin folders of your WordPress installation with the settings set to
max_execution_time = 0 memory_limit = 150M
If this does not work then you have to edit the value in your php.ini file. The location of this file depends on your operating system and a quick Google search will yield plenty of literature on how this is done.