I actually think @chrillek’s point was completely on-topic. Since…
…I assume that was a warning from DT3.
Your license is not tied to databases, but to the application.
I haven’t upgraded yet, but I guess the license code changes when you upgrade your license. You would have to enter your new license key in DT3.
But why use DT3 if you have upgraded to DT4?
Edit: the FAQ says:
When I upgrade to DEVONthink 4, do I trade-in my license?
No, all license codes for DEVONthink 3 remain valid and you do not need a new, separate license for DEVONthink 4.
– so perhaps your DT3 version is older than 3.9.11.
In any case, as long as you have a valid license, you should be able to open your databases.