Today I have added an alternative and newer version of PHP 5.5 to my server and immediately I find I can install BackdropCMS.
My VPS, using Plesk 12.0.18, had been provided with PHP 5.5.9 and this has proved satisfactory for several Drupal 7 sites, so it was a setback to be unable to install Backdrop yesterday.
I had no idea how to track the problem - I could find no error reports within my server logs, but this may well be lack of knowledge on my part.
So I had the idea of installing an alternative version of PHP, which is a non-trivial and fairly lengthy task but one I had tackled a year ago, and fortunately I had kept notes of the steps to be taken.
As soon as I set Plesk to use this newer version (PHP 5.5.20) for this domain, Backdrop installed smoothly and successfully.
I have now tried to identify the problem and it would seem to be the version of mysql libraries installed within PHP. The ones labelled 'Client API version 5.5.40' were preventing the installation of one of the Backdrop data tables, namely 'watchdog'.