While I agree with everything scalabilitysolved has said, there is an interesting feature in the offing for Postgres, scheduled for the 9.4 Postgres release, namely, jsonb: http://www.postgresql.org/docs/devel/static/datatype-json.html with some interesting indexing and query possibilities. I mention this as you tagged Mongodb and Couchbase, both of which use JSON (well, technically BSON in Mongodb's case).
Of course, querying, sharding, replication, ACID guarantees, etc will still be totally different between Postgres (or any other traditional RDBMS) and any document-based NoSQL solution and migrations between any two RDBMS tends to be quite painful, let alone between an RDBMS and a NoSQL data store. However, jsonb looks quite promising as a potential half-way house between two of the major paradigms of data storage.
On the other hand, every release of MongoDB brings enhancements to the aggregation pipeline, which is certainly something that seems to appeal to people used to the flexibility that SQL offers and less "alien" than distributed map/reduce jobs. So, it seems reasonable to conclude that there will continue to be cross pollination.
See Explanation of JSONB introduced by PostgreSQL for some further insights into jsonb.