Can SQL Server Bulk Copy cause timeouts when querying the table
It definitely could.

It requires locks just like any other insert operation. If enough locks are taken it will escalate to a full table lock (assuming the table allows it). Any insert operation like this would block anything else that was trying to read the data, unless NOLOCK was specified for those queries (which I am not recommending here).

You can actually set up a trace to see what types of locks are being taken if you are interested. Just set up a Profiler trace that gets the "Locks: Acquired" and "Locks: Escalation", and then do a bulk insert.

Word of warning, Profiler can cause a hit to performance, so I would either run the trace on a dev server, or if you have to run it in production, export the script and execute it as a server-side trace.