1) You need an embedded relational database with a small foot print. Here PostgreSQL cannot compete with SQLite.
2) You have an application which does not support PostgreSQL.
3) You have petabytes worth of data. Here you want to look into something like Greenplum, a PostgreSQL fork.
Citus and other community extensions can help if you want to go distributed and fault tolerant
1) You need an embedded relational database with a small foot print. Here PostgreSQL cannot compete with SQLite.
2) You have an application which does not support PostgreSQL.
3) You have petabytes worth of data. Here you want to look into something like Greenplum, a PostgreSQL fork.