Does a computer assembled only from CE marked subassemblies still need CE testing?
Yes, the completed system needs independent testing and labeling to be sold in the European Union. I'm not an agency expert but all the products I've been associated with had to be tested with all the components connected that would be in the final system. 

The CE mark certifies the system meets a variety of requirements, not the least of which is electrical safety.  If you take five components that all have a CE mark but just barely, the sum of the five may push the system outside of limits. Let's say that the components all have 50uA leakage levels and all pass.  The assembled system might have something like 150uA to 250uA leakage and over the 100uA limit.  

The same for RF emissions levels, dielectric breakdown, etc.

The good news is that with CE marks components, you know that they were safety tested and passed and chances are good that the system will, as well.

Good luck!