Sometimes referential integrity seems not to work

In a database I am working, I sometimes have to delete all the records in
some tables. According to the referential integrity defined in the creation
of the tables, postmaster should not delete the records, but it does. I have
used the following commands: "delete from table_1" and "truncate table_1".
When deleting records individually, referential integrity works (postmaster
does not allows deletion of records which are referred by another table).
Is there some difference between deleting records individually or at once?
Is there any possibility of the user who does this deletion (postgres, normal
users) affects the behaviour of referential integrity in the tables? That is,
can the postgres user delete records despite referential integrity?

Thanks.