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No, the interval type supports reduced precision but none of the other date/time types do.

Postgres allows you to roll your own with create type but unfortunately wont allow contraints to be added to the type which limits it's usefulness in this scenario. The best I can come up with requires you to repeat check constraints on every field where the fuzzy type is used:

create type preciseness as enum('day', 'month', 'year');
create type fuzzytimestamptz as (ts timestamptz, p preciseness);
create table t( id serial primary key,
                fuzzy fuzzytimestamptz
                    check( (fuzzy).ts is not null 
                           or ((fuzzy).ts is null and (fuzzy).p is not null) ),
                    check((fuzzy).ts=date_trunc('year', (fuzzy).ts) or (fuzzy).p<'year'),
                    check((fuzzy).ts=date_trunc('month', (fuzzy).ts) or (fuzzy).p<'month'),
                    check((fuzzy).ts=date_trunc('day', (fuzzy).ts) or (fuzzy).p<'day') );

insert into t(fuzzy) values (row(date_trunc('year', current_timestamp), 'year'));
insert into t(fuzzy) values (row(date_trunc('month', current_timestamp), 'month'));
insert into t(fuzzy) values (row(date_trunc('day', current_timestamp), 'day'));

select * from t;

 id |              fuzzy
----+----------------------------------
  1 | ("2011-01-01 00:00:00+00",year)
  2 | ("2011-09-01 00:00:00+01",month)
  3 | ("2011-09-23 00:00:00+01",day)

--edit - an example equality operator:

create function fuzzytimestamptz_equality(fuzzytimestamptz, fuzzytimestamptz)
                returns boolean language plpgsql immutable as $$
begin
  return ($1.ts, $1.ts+coalesce('1 '||$1.p, '0')::interval)
         overlaps ($2.ts, $2.ts+coalesce('1 '||$2.p, '0')::interval);
end;$$;
--
create operator = ( procedure=fuzzytimestamptz_equality, 
                    leftarg=fuzzytimestamptz, 
                    rightarg=fuzzytimestamptz );

sample query:

select *, fuzzy=row(statement_timestamp(), null)::fuzzytimestamptz as equals_now,
          fuzzy=row(statement_timestamp()+'1 day'::interval, null)::fuzzytimestamptz as equals_tomorrow,
          fuzzy=row(date_trunc('month', statement_timestamp()), 'month')::fuzzytimestamptz as equals_fuzzymonth,
          fuzzy=row(date_trunc('month', statement_timestamp()+'1 month'::interval), 'month')::fuzzytimestamptz as equals_fuzzynextmonth
from t;
 id |               fuzzy                | equals_now | equals_tomorrow | equals_fuzzymonth | equals_fuzzynextmonth
----+------------------------------------+------------+-----------------+-------------------+-----------------------
  1 | ("2011-01-01 00:00:00+00",year)    | t          | t               | t                 | t
  2 | ("2011-09-01 00:00:00+01",month)   | t          | t               | t                 | f
  3 | ("2011-09-24 00:00:00+01",day)     | t          | f               | t                 | f
  4 | ("2011-09-24 11:45:23.810589+01",) | f          | f               | t                 | f