You are seeing many of your Office documents opening up from SharePoint 2013 in a "Read-Only" state, but other files are opening in an "Online" state. This is sometimes shown as a hard limit in the size of the document being opened.
Example: Documents over 6MB in size are opening as "Read-Only", but documents smaller than that are opening normally.
It is possible the FileWriteChunkSize property has been changed in the SharePoint 2013 Farm
1. Check the property size in SharePoint Management shell(Specify your own Web Application URL):
$webapp = Get-SPWebApplication http://sp2013
$webapp.WebService.FileWriteChunkSize | Write-Host
2. If the value shown is not equal to 64320 then proceed to step 3.
3. Reset to default value of 64320 in SharePoint Management shell:
$webapp.WebService.FileWriteChunkSize = 64320
SPWebApplication.WebService.FileWriteChunkSize shouldn't be modified from the original value. The value is used in the chunking algorithm and doesn't act as one would think, for example setting that value to be 1 GB doesn't cause the chunks to be 1 GB each and also doesn't cause SharePoint to start chunking at that 1 GB. The value should be left at its original state and not modified.