No, that's not possible. At least, not in an easy and user friendly way. It's actually never been possible. You're probably thinking of the Gnome Panel that was used by default in the past. It's a different program, and though they look similar, they're rather different.
I understand your use case, and it might be nice to be able to reorganize the indicators. One thing that needs to be considered in this particular case, is that the weather indicator is an "app", whereas the message, network, sound and calendar (datetime) are all system indicators. I don't know if they should be mixed, since that would create inconsistencies.
For now, the answer is no, you can't. I'll bring it up for discussion in the Unity design list, and update this answer when it's been discussed.