Battery?

Have you verified your battery is discharged??  When you hook 2 different chargers to your battery and then hook a new battery to the charger and get the same results, not knowing what is going on, I make the assumption that you're fully charged and the chargers are cycling in a "maintenance" or trickle charge.  Since your charger has a 6V setting, we can rule out incompatibility.

You should have a volt meter and a hydrometer (allows you to check fluid in each cell)! 

I've often observed bubbling in one cell only (or all bubble and one doesn't) when charging to identify a bad cell.  This isn't fool proof and can be dangerous as all acid batteries produce Hydrogen and can explode and spray acid all over the place (I've had 2 batteries explode on me -both times it was a short inside the case).  It's painful and unhealthy.  So, using a hydrometer to check the specific gravity is much, much safer.  Google for info including a stock number for one at your local auto parts store.

The voltmeter will tell you if you have a discharged battery.  Remember that a fully charged battery will read higher than its rated voltage, i.e. 6.3-6.5 for a 6V battery.

If your battery is fully charged, the chargers are very possibly functioning correctly.  If your battery is low, i.e. below 6.25, then one of two things is happening:  Both chargers are simultaneously not working or your battery and the new battery won't take a charge.  Since you've used 2 different chargers, I'd assume they work.  Since you've hooked up a new battery, I can also assume (at least for the new battery) that the batteries are probably ok.  Testing the battery will confirm or rule out bad.

And now, just in case....

The most overlooked problem (IMHO) I've seen are the cables.  When's the last time you inspected and cleaned?  Have your cable ends ever been replaced?  It's easier to inspect those as you can actually see the wire.  You're looking for a white powdery build up that literally acts as an insulator.  I use a wire bristle post cleaner but you can use a knife or even a screw driver - anything to scrape off the contact surfaces of both the posts and terminals.  If there is swelling or visible powder under the insulation on the wires, you may need to replace them.  Remember that 6V needs a larger gauge than 12V wires (1/0 IIRC).  You can sometimes repair cable ends by dunking in a saturated solution of water/baking soda and let it fizz.  If it's under the insulation, you'll have to slit it open far enough to see clean wire and neutralize all the corrosion.  I always try to disassemble when cleaning like this - lets the solution get into all the nooks/crannies.  If it goes deep, you may be looking at all the cables from battery down to your starter (that's at the extreme end).

A word of caution.  I've seen batteries destroyed by sprinkling baking soda on top to control the acid buildup.  Likewise, pouring the baking soda solution over a battery will do the same.  If there's a buildup on the battery case, I dip, squeeze (a paper towel) and then carefully wipe the battery down.

This is way too much information but hopefully it helps.