Master_Xan

What do to when a planet REFUSES to fall to your all-powerful hand:
 
1) Sab the shields. If that fails, sab the troops THEN get the shields. If you can't take down the shields, it is unlikely you'll be able to capture a general, so try if you want, but don't expect much.
 
2) When sab fails, run a few espionage missions. You will often find hordes of people you didn't know were there, such as generals, spec ops forces, ect. This gives you a good idea of what you're up against, but doesn't help you take the planet directly.
 
3) Bombard the planet with a huge fleet. Use Y-Wings, B-Wings, and heavy-duty warships. This will overwhelm the shields, often resulting in heavy troop causalties, with further bombardments taking down the shield generators themselves.
 
4) If a sufficiently large fleet isn't available, or you're worried about taking losses from laser batteries, and they only have one shield generator you can still assault the planet directly with troops. Get some troopers with high attack ratings, like Wookies, and set a General in orbit with the fleet for best results.
 
5) If you can't sab them, can bombard them, and can't assault them into submission, there are still options. If you're Imperial, use a Death Star. Since you're not Imperial, you can *try* and do incite uprising missions, but this will almost certainly fail. Your next option, as the Alliance, is to...
 
6) Leave. You heard me, leave. If possible, do things to other planets in the sector which will impact the offending planet's political alignment in your favor. If not, then do whatever you can to draw them off the planet. If you're blockading the planet, they will never leave, so you should take out troops elsewhere (to make them send troops from the offending planet to others 'in need'). This will make it easier to run sab missions.
 
7) Your last hope is to destroy the rest of the players economy, making them loose from low maint. points. Or to simply avoid the planet altogether, which is a bad idea.
 
Hope that helps...

If you use RebEd, you can see which ships have higher bombardment ratings than others. Which has higher value, ISD II or Victory SD II? Check RebEd. Also helpful is having large numbers of fighters with bombardment ratings; if I recall correctly, Y-wings and B-wings both provide bombardment strength for the Alliance.
 
You should note from this that not all ships have bombardment capability, and so just having lots of ships isn't enough. Ex. Corellian Gunships, if I recall correctly, don't have any bombardment strength at all. That's why you should look and see which ships have better capabilities. They must have at least 1 to be of use, though obviously higher numbers are better.
 
In the situation you described, I would suggest bringing in a group of fighter carriers. They are cheap to make, and arming them with, say, B-wings, is much quicker than making dozens of capital ships. Bombard the planet with whatever ships either carry fighters with bombardment or have bombardment themselves; don't have troop transports in the fleet, or they may be hit by the LNR guns (costing their compliment of troops). If you're in the game and don't remember which ships have bombardment, separate a ship of one class from the fleet and see if the game allows you to bombard with that ship. If yes, then that class of ship has at least a 1 in bombardment. If no, then it won't help you take down any shields.
 
The game may list bombardment in the Encyclopedia files; I can't remember for sure. I don't think it does, but maybe...
 
Also, if you can sabotage the shields before your opponent has nothing left but that one planet, it will be easier. In that case, Vader may be elsewhere, other defenses may be off-planet, etc. The computer doesn't always replace a lost shield gen with another one. Conversely, if you can sabotage the LNRs, that makes it much safer to keep trying to take down the shields. Of course, you can't always get them out even then, especially on Coruscant where defenses are often high throughout the entire game.
 
I may also mention that even if your fleet bombards and the shields hold, they may not hold if you bombard again. I usually try a few times before I decide they really can't break through. This may be painful if they have LNR stations, but at the end of the game, you can replace the ships if you really have to.