In theory if the alternator is working properly and all connections are solid then pulling a battery cable should not make a difference - but pulling it with a marginal alternator or bad connections in the system could spike and fry your engine computer and God knows what else. The battery acts like a surge suppressor.
There's one thing wrong with focusnewbie's testing explanation. If you have low system voltage at the battery terminals it is not necessarily a bad battery. The alternator could be undercharging the battery.
A good car alternator should supply 13.8v at minimum at the battery terminals at idle speed with the headlights on. Normally a battery without the engine running will go to 12v. Of course, you need a good battery on the terminals when measuring this. This does make it a bit difficult to troubleshoot a charging/starting electrical problem in a car system, in determining if it's really the battery or not.
For that reason I find it best to start by troubleshooting the battery out of the car. That is actually pretty easy, you need a car battery charger and a battery load tester.
The best battery charger for a car battery is a 2/6 amp charger, not the 10/2/50 charger/starter charger. The 2/6 are often sold as motorcycle battery chargers but they are actually better car battery chargers than the more powerful ones. The reason why is that the slower charge rate of a weaker charger (which of course takes more time to recharge the battery) can help to dissolve light sulphation on the battery plates better than a high speed high current "quick charge" (at least, this has been my observation - YMMV) As for the 12v battery load tester there are 2 types, a "carbon pile" that can take up to 500amps of current from the battery, and a lighter duty "100 amp" load tester.
The process to test the battery is pretty simple. First, charge it overnight with the battery charger. When you first plug in the charger observe the current meter on the charger. A car battery that is seriously discharged yet still a good battery should be able to pull from 6amps to more from the charger. As the battery gets more charged up, the charge current will drop. Anyway, after you have given it a good 12 hours on the charger, when it's fully charged then disconnect it from the charger for a couple hours, then test with a voltmeter - it should read 12v. Then at that time use a battery load tester on it (I use the cheapo Harbor Freight 100 amp load tester it has a voltmeter in it) and give it a load test. Then for extra good measure, let it sit for a couple weeks (or at least over the weekend) then hit it with the load tester again and make sure it's still able to supply 100 amps and it's still reading 12v.
The Harbor Freight load tester is only $22 it's definitely worth it.
Typically car batteries in normal car electrical systems fail 3 ways:
1) shorted cell or cells - this is usually manufacturing defect or age or sometimes impact damage (bouncing around), and battery voltage will go below 12v
2) sulphation - this is a normal process in lead acid car batteries and is unavoidable - symptom is battery will hold charge, will show 12v - but 15-30 seconds on 100amp load tester and voltage on the battery will start to fall. It's much easier to see using a carbon pile that will pull 500 amps off the battery.
This is why getting the highest CCA battery that will fit is one way to lengthen battery life - a 3 year old battery may be down to 25% of original capacity due to sulphation - but if the original called for a 300CCA battery and you put an 800 cca battery in there, you will still have 200 cca left even when it's down to 25%. Also, a higher capacity battery does not discharge as much (as a percentage) when starting.
All lead acid batteries sulphate. Sulphur is deposited on the plates as the battery discharges (and the electrolyte changes from sulphuric acid to water) then when the battery is recharged most of the sulphur is dissolved as the electrolyte changes back to sulphuric acid. But there is a microscopic amount that never dissolves again and over 4-6 years it builds up until the battery capacity is destroyed.
3) high leakage - symptom is battery will take a charge, display 12v, pass a load test - but then 24 hours later will fail the same load test (or barely pass it) This is almost always an age issue, it is caused by lead being shed from the plates collecting at the bottom of the battery until it creates an electrical path for the battery to self-discharge.
Now, if the alternator is bad, then it rapidly hastens the death of the car battery through 2 ways. First, if the alternator is putting out too high a charge voltage, once the battery has reached the stage of full charge, it starts decomposing the water in the electrolyte (turning it into hydrogen and oxygen) and the water level starts dropping. Too much charge current can also heat the battery and cause the plates to warp. If the battery is run with low electrolyte for a long time then the sulphur will deposit out onto the plates a lot faster. Second if the alternator is not charging the battery enough, then the battery can be undercharged, which will rapidly hasten sulphation.
Some people put a lot of stock in measuring the specific density of the electrolyte using a battery hydrometer. In my experience these are useless most of the time. First, the ones sold at the auto store with the colored balls are very inaccurate. Second, a hydrometer does not work if some of the electrolyte has decomposed and the water level has dropped - since that process changes the specific gravity of the electrolyte.
The last thing about car batteries - they store a huge amount of power. If you drop a wrench or screwdriver across a battery terminal set you will likely weld it on, and the subsequent rapid discharge will heat the battery so much it can cause it to spray out sulphuric acid, or even explode. So be careful.
