Yes, your motherboard should be able to support 6 GB of RAM as long as you have DDR3 1066/1333. Here you can get more information about http://www.intel.com/content/www/us/en/support/boards-and-kits/desktop-boards/000005876.html System Memory for the Intel® Desktop Board DH55HC.
Your board has four sockets for installing memory DIMMs. Two of these sockets are for DIMMs that will be connected to the chipset's first memory channel. The other two are for the chipset's second memory channel. At present, I will presume that you have two 1GB DIMMs in the first and third sockets (or the second and fourth; namely to have one connected to each memory channel [Aside: so that interleaved accesses can occur]). You need to move one of these DIMMs to a different socket so that the two DIMMs are connected to the same memory channel. Then, you can install the new DIMMs in the sockets for the other memory channel.
Overall, the point I am trying to make is that the DIMMs connected to a memory channel need to closely match (i.e. be the same size and same speed)...
Hope this helps,