Yes, and no.
It depends on the architecture of your application. We'll assume that you're building an n-tier application, since the vast majority of applications these days tend to follow that model.
Validation in the user interface is designed to provide immediate feedback to the end-user of the system to prevent functionality in the lower tiers from ever executing in the first place with invalid inputs. For example, you wouldn't even want to try to contact the Active Directory server without both a user name and a password to attempt authentication. Validation at this point saves you processing time involved in instantiating an object, setting it up, and making an unnecessary round trip to the server to learn something that you could easily tell through simple data inspection.
Validation in your class libraries is another story. Here, you're validating business rules. While it can be argued that validation in the user interface and validation in the class libraries are the same, I would tend to disagree. Business rule validation tends to be far more complex. Your rules in this case may be more nuanced, and may detect things that cannot be gleaned through the user interface. For example, you may enforce a rule that states that the user may execute a method only after all of a class's properties have been properly initialized, and only if the user is a member of a specific user group. Or, you may specify that an object may be modified only if it has not been modified within the last twenty-four hours. Or, you may simply specify that a string value cannot be null or empty.
In my mind, however, properly designed software uses a common mechanism to enforce DRY (if possible) from both the UI and the class library. In most cases, it is possible. (In many cases, the code is so trivial, it's not worth it.)