If anyone else has this issue, check to see which OPC client and which OPC specification you are using.
For our OPC connections, we *were* using FactoryTalk Gateway, which uses the OPC-DA 2.05A specification. This gave us repeated intermittent "stuck tag" problems of the sort described above. Upgrading PCo did not help (contrary to what I erroneously said above), nor did anything else.
That is, nothing helped until we switched over to KepServer EX 5, which uses the OPC-DA 3.00 specification. Since then, we have not had any problems.
Therefore, if my experience is any indication, if you plan to use PCo, I would strongly advise against using any OPC-DA 2.05A client, especially FactoryTalk Gateway.
I want to sincerely thank Steve Stubbs for recommending KepServer EX and for all the rest of his help over the course of many months. My company's whole manufacturing department is very grateful.
Greg