Today I got the reply from the BoldGrid support of the W3 Total Cache plugin.
The explanation to this occurance is that there was a temporary problem with the directory while the backup ran through. It seems that the directory was moved or modified with UpdraftPlus while W3 Total Cache tried to open the directory at the same time. This let W3TC throw this continuous error. The backup was not necessarily affected by this though.
Given the fact that this situation was a temporary inconvinience, there is nothing really to fix for now and I call this case closed.
