Yes of course, I'm sure you've heard this before: "Bitcoin's transaction aren't anonymous but rather pseudo-anonymous[/i]."

Coinbase is a centralized wallet and exchange and can be easily analyzed using any blockchain analysis tool since they "consolidate" most of the inputs using multiple and different (deposit) addresses, using addresses' inputs together will link them to the same wallet used by any one of those addresses.
Although using that tool (link above) is as easy as checking a transaction/address in a blockexplorer, a normal internet guy might find it hard but not the authorities with sophisticated tools and agents, they can always find a way.

And in addition, your KYC data is always available for the Gov't and Coinbase to see and every deposit address you've generated will be marked as "yours".

Next time, post questions like this in the Bitcoin Technical Support board in order to get more accurate answers.