public static void main(String[] args) {

    String url = "jdbc:postgresql://localhost:5432/testdb";
    String user = "user12";
    String password = "34klq*";

    try (Connection con = DriverManager.getConnection(url, user, password);
            Statement st = con.createStatement();
            ResultSet rs = st.executeQuery("SELECT VERSION()")) {

        if (rs.next()) {
            System.out.println(rs.getString(1));
        }

    } catch (SQLException ex) {

        System.out.println("Error: "+ ex);
        Logger lgr = Logger.getLogger(PostgresCommand.class.getName());
        lgr.log(Level.SEVERE, ex.getMessage(), ex);
    }
}

abr 06, 2018 8:28:04 AM org.postgresql.core.v3.ConnectionFactoryImpl log
Error: org.postgresql.util.PSQLException: FATAL: no existe la base de datos «testdb» (pgjdbc: autodetected server-encoding to be ISO-8859-1, if the message is not readable, please check database logs and/or host, port, dbname, user, password, pg_hba.conf)
ADVERTENCIA: SQLException occurred while connecting to localhost:5432
org.postgresql.util.PSQLException: FATAL: no existe la base de datos «testdb» (pgjdbc: autodetected server-encoding to be ISO-8859-1, if the message is not readable, please check database logs and/or host, port, dbname, user, password, pg_hba.conf)
    at org.postgresql.core.v3.QueryExecutorImpl.receiveErrorResponse(QueryExecutorImpl.java:2433)
    at org.postgresql.core.v3.QueryExecutorImpl.readStartupMessages(QueryExecutorImpl.java:2566)
    at org.postgresql.core.v3.QueryExecutorImpl.<init>(QueryExecutorImpl.java:131)
    at org.postgresql.core.v3.ConnectionFactoryImpl.openConnectionImpl(ConnectionFactoryImpl.java:210)
    at org.postgresql.core.ConnectionFactory.openConnection(ConnectionFactory.java:49)
    at org.postgresql.jdbc.PgConnection.<init>(PgConnection.java:195)
    at org.postgresql.Driver.makeConnection(Driver.java:452)
    at org.postgresql.Driver.connect(Driver.java:254)
    at java.sql.DriverManager.getConnection(DriverManager.java:664)
    at java.sql.DriverManager.getConnection(DriverManager.java:247)
    at database.PostgresCommand.main(PostgresCommand.java:33)

abr 06, 2018 8:28:04 AM org.postgresql.Driver connect
GRAVE: Connection error:
org.postgresql.util.PSQLException: FATAL: no existe la base de datos «testdb» (pgjdbc: autodetected server-encoding to be ISO-8859-1, if the message is not readable, please check database logs and/or host, port, dbname, user, password, pg_hba.conf)
    at org.postgresql.core.v3.QueryExecutorImpl.receiveErrorResponse(QueryExecutorImpl.java:2433)
    at org.postgresql.core.v3.QueryExecutorImpl.readStartupMessages(QueryExecutorImpl.java:2566)
    at org.postgresql.core.v3.QueryExecutorImpl.<init>(QueryExecutorImpl.java:131)
    at org.postgresql.core.v3.ConnectionFactoryImpl.openConnectionImpl(ConnectionFactoryImpl.java:210)
    at org.postgresql.core.ConnectionFactory.openConnection(ConnectionFactory.java:49)
    at org.postgresql.jdbc.PgConnection.<init>(PgConnection.java:195)
    at org.postgresql.Driver.makeConnection(Driver.java:452)
    at org.postgresql.Driver.connect(Driver.java:254)
    at java.sql.DriverManager.getConnection(DriverManager.java:664)
    at java.sql.DriverManager.getConnection(DriverManager.java:247)
    at database.PostgresCommand.main(PostgresCommand.java:33)

abr 06, 2018 8:28:04 AM database.PostgresCommand main
GRAVE: FATAL: no existe la base de datos «testdb» (pgjdbc: autodetected server-encoding to be ISO-8859-1, if the message is not readable, please check database logs and/or host, port, dbname, user, password, pg_hba.conf)
org.postgresql.util.PSQLException: FATAL: no existe la base de datos «testdb» (pgjdbc: autodetected server-encoding to be ISO-8859-1, if the message is not readable, please check database logs and/or host, port, dbname, user, password, pg_hba.conf)
    at org.postgresql.core.v3.QueryExecutorImpl.receiveErrorResponse(QueryExecutorImpl.java:2433)
    at org.postgresql.core.v3.QueryExecutorImpl.readStartupMessages(QueryExecutorImpl.java:2566)
    at org.postgresql.core.v3.QueryExecutorImpl.<init>(QueryExecutorImpl.java:131)
    at org.postgresql.core.v3.ConnectionFactoryImpl.openConnectionImpl(ConnectionFactoryImpl.java:210)
    at org.postgresql.core.ConnectionFactory.openConnection(ConnectionFactory.java:49)
    at org.postgresql.jdbc.PgConnection.<init>(PgConnection.java:195)
    at org.postgresql.Driver.makeConnection(Driver.java:452)
    at org.postgresql.Driver.connect(Driver.java:254)
    at java.sql.DriverManager.getConnection(DriverManager.java:664)
    at java.sql.DriverManager.getConnection(DriverManager.java:247)
    at database.PostgresCommand.main(PostgresCommand.java:33)

try{
...
}catch(Exception e)
{
//Puedes investigar los métodos de e para obtener información
}