...

I am curious to know if its possible to have a router with its two interfaces, say configured as, 1.1.1.1/16 and 1.1.1.2/16. Theoretically, i see nothing which can stop a router from doing this.

Cisco's don't let you do this. I have always considered that broken, although I'm sure Cisco thinks it's a feature.

bridge irb inte f5/0 no ip add bridge group 1 inte f5/1 no ip add bridge group 1 inte bvi 1 ip add 1.1.1.1 255.255.0.0 bridge 1 prot ieee bridge 1 route ip bridge 1 bridge ip