Try this.. create a string variable. This way you can trouble shoot the variable before assigning it to the hyperlink.
Code:
Dim strPath as string
strPath = "\\***private***\engineering_data\***Private***\Design\Fitout\Label Plate\_LATEST REV" & Me.TextboxName & ".pdf"
debug.print strPath
The debug line will print your string variable in the Immediate Window. You can view the immediate window using Ctrl+G on your keyboard. There you can study the hyperlink and see where there may need to be an adjustment. You can try pasting it in Windows Explorer. You could also paste it here and maybe we can see something.