If the max number exceeds the parameter value, new connection will not be able to connect to the database. Grid Control will report the connection problem and the database will give error in trace file:
ORA-00020: maximum number of processes (300) exceeded
In that case, the issues can be confirmed by number of OS sessions and by database view v$process:
$ echo $ORACLE_SID
$ ps -ef | grep $ORACLE_SID | wc -l
299
SQL> select count(*) from v$process;
COUNT(*)
----------
297
To increase the value, run an alter statement. Then a database bounce is necessary.
SQL> alter system set processes=500 scope=pfile;
I read that below three parameters are related by a formula (in 10G, but might not be always). If database reports ORA-00018 error, you may also consider modifying parameter PROCESSES (or find why the number of sessions jumped).
ORA-00018: maximum number of sessions exceeded
processes=x
sessions=x*1.1+5
transactions=sessions*1.1