How do I recover from an unchecked exception?

try {
    throw new IllegalArgumentException();
} catch (Exception e) {
    System.out.println("boom");
}

// insert the user
try {
   hibernateSession.save(user);
} catch (ConstraintViolationException e) {
   throw new MessageException("Username " + user.getName() + " already exists. Please choose a different name.");
}

try {
   ... render the page
} catch (MessageException e) {
   ... render a nice page with the message
} catch (Exception e) {
   ... render "we logged your problem but for now you're hosed" message
}