How can I use Views to create a page with 3 Content Types, grouped by type, limited to x number of each type?
Here's a solution using only the Views UI.


Create a view with the following master settings: display 3 items, sort post date descending with the node tile included in your fields. Probably as you have it already.
Add a page or block display with type == News articles. This will be your parent view display. Again, you might have this already, but your type filter may need to be adjusted. We just want 3 news articles here.
Add an attachment display with title as Events and update type == events. Attach this to your parent view display above.
Add another attachment display with title as Press Releases and update type == press releases. Attach this to the parent view in the second bullet point above.


You can find the attachment to setting in the second center column on the views ui, assuming this Drupal 7.

Jason.