Absolutely. Google Bikini Atoll. Several battleships were (moored) targets in the Operation Crossroads test. The bombs (one air burst, one shallow water burst) ruptured weld seams, started fires, and so on. A close enough burst will disable the crew, too, so that damage control would be impaired.
Guided missiles were appearing at the end of WW2:
The Germans developed and used the “fritz” FX1400 glider bomb (radio controlled from a nearby bomber) to sink the Italian BB Roma, and nearly sank two other BB’s (Italian Italia and British Warspite).
The USN experimented firing “liberated” German V-1 and V-2 rockets from subs and surface ships. (The US eventually developed it’s own missiles based on these pioneering tests.)
The USS Missippi (finished in 1917) was used in the 50’s as a missile test ship.
Submarines were also becoming more deadly as the war dragged on: Snorkles (able to operate submerged more effectively), more powerful torpedo warheads (torpex explosives for example), radar. Eventually, submarines were able to operate just as fast submerged as they did on the surface.
While the aircraft did speed the end of the battleship, the guided missile seems to have sealed the deal…
Even during the age of the battleship, many naval architects kept trying to build the ultimate BB. Armored enough to defeat the biggest guns in service, but still have enough useful displacement left over for engines and weapons. Then some yutz builds a bigger gun.
Another consideration is the support infrastructure. The designers had to deal with the restrictions of maximum draft (to prevent grounding while pulling into harbors), hull width and/or length (drydock dimensions, Panama Canal dimensions, etc.). With a given max hull size, max draft, and max available space for engines (based on current tech, which you can then get a close estimate of max possible engine power), desired speed (longer hull for a given engine HP usually gives more speed), amount of area you want protected by armor, etc, etc, etc… quite a juggling act.
Eventually, I think that the armored warship would have faded away, as it appears that missile and torpedo power kept getting better, and a practical limit in armor (weight and cost) means that defense needs to focus on “avoid being hit”, as opposed to thicker and thicker armor.
Todays USN DD’s, the Arleigh Burke class, are 8000 tons. Almost as much displacement as a lot of Heavy Cruisers that fought in WW2. The aegis guided missile cruisers, the Ticonderoga class, are not much bigger. (9500 tons)
With limited peacetime budgets, ships need to be able to do multiple missions (ASW, AA, Anti-surf, littoral warfare.), and this accounts for the growth in the size of the USN DD’s.
In a shit-hit-the-fan war, cheaper, smaller, easier to build mission specific ships would be churned out, if their is enough time. (Like the USN DE types of WW2. Max 25 knots, small hull. Useful for ASW/convoy escort jobs, but not as useful for fast combat TF’s or air defense.)