SQL select elements where sum of field is less than N

Given that I've got a table with the following, very simple content:

# select * from messages;
  id | verbosity 
 ----+-----------
   1 |        20
   2 |        20
   3 |        20
   4 |        30
   5 |       100
 (5 rows)

I would like to select N messages, which sum of verbosity is lower than Y (for testing purposes let's say it should be 70, then correct results will be messages with id 1,2,3). It's really important to me, that solution should be database independent (it should work at least on Postgres and SQLite).

I was trying with something like:

SELECT * FROM messages GROUP BY id HAVING SUM(verbosity) < 70;

However it doesn't seem to work as expected, because it doesn't actually sum all values from verbosity column.

I would be very grateful for any hints/help.