Could there be a possibility that some Romans knew that there was a country north east of India? Because as far as I know the Romans traded with Indians.
I started to deliver some of my trademark withering sarcasm, but it turns out I had to eat crow. I knew that the answer was "yes", and I assumed that a trivial google search would confirm my knowledge. It took 3 adjustments to the query and 3 pages of results before I found the above.
Here are some other sources (turns out my first search was grossly flawed)
They did know.
Roman knowledge of China is attested in the Geographia, work of the famous Claudius Ptolemy. Which is not to say, however, that the Romans knew much at all about the Han Empire (or vice versa, for that matter). For instance, Ptolemy's map of the Far East coastline is rather distorted:
In Chinese records, the Han Emperor first received Roman emissaries in 166 A.D.