I hadn't wanted to burden forum members with the background to my current problem, but will explain now. Thanks for your advice which I have been following. In the meantime thank you cdhw for to efforts to solve my problem. If have any success there I will come back and add an update to this forum in case anyone else experiences a similar problem.
So, based on the fact that Numbers and the earlier version of Excel were able to open the non-corrupted files, I think it may be an issue related to the latest (Ver 15.18) of Excel that is the problem, so I will give the Microsoft Mac Users Forum a try. I did download Text wrangler but to be honest I am not sure what I was looking at.
EXCEL FOR MAC GRANT ACCESS INSTALL
I currently don't have LibreOffice installed on my MBP, but given that Numbers and the older version of Excel could open the (non-corrupted) files, I think that if I were to download and install LibreOffice it would also be able to open these same files. Similarly if I right click and select Numbers (Ver 3.6.1) the file opens okay. However, if I right click on the same file I can select the older version of Excel (Ver 14.6.0) and the file opens. If I double click the files that appear okay in, then my default version of Excel (Ver 15.18) results in the the 'Grant Access' and ' Cannot open file' windows I included in my original post.
Using and I can see a thumbnail of each file, so the ones that appear as a blank thumbnail are obviously corrupted. I have checked a number of the xlsx and xls files and I have Read & Write privileges in each case. An update and some comments on your suggestions.