If you add some debbuging statements, do things proceed through your ifs properly?

if( $purchases >= 50 && ! in_array( 'bronze_user', $user->roles ) ) {
        // Remove role
        $user->remove_role( 'subscriber' );

        // Add role
        $user->add_role( 'bronze_user' );
    }
 if( $purchases >= 100 && ! in_array( 'silver_user', $user->roles ) ) {
        // Remove role
        $user->remove_role( 'bronze_user' );

        // Add role
        $user->add_role( 'silver_user' );
    }
 if( $purchases >= 200 && ! in_array( 'gold_user', $user->roles ) ) {
        // Remove role
        $user->remove_role( 'silver_user' );

        // Add role
        $user->add_role( 'gold_user' );
    }

Hope it does helps.