No, you aren't doing anything wrong. That is what Palistine lime tastes like. I have one too and that is what they taste like. The only thing sweet about it is the name. They are insipidly boring.
You might want to try a graft, since you already have good roots there. Also I recomend the thornless Mexican Lime since the standard variety has LOTS of thorns.
If you don't want to try a graft, just be sure to plant the next tree a little to one side or the other, not in exactly the same hole as the first tree.

Thanks softmentor for the green light to rip that baby out!
I have far too little space (far too many plants I want) to have a plant that is insipidly boring.

Out of curiosity.. why not use the same hole?

Thanks, again
Kate

It's not recommended to use the same exact spot because a lot of time the previous "resident" will have left behind some factors that will inhibit growth. and if planting the same family of plant, you will be very susceptible to root problems as the previous plant roots decompose.

Are you using them while green or yellow? I'm not growing any sweet limes, but I do buy them at a local farmers market. They are always yellow ripe and are very good that way. Never seen them sold while green nor have I tried them as such. Not sure if that helps of not.

I wouldn't rip it out. Just graft varieties that you like better onto it. No sense wasting a good root system. Grafting is easy to do. I would try find someone near you willing to give you some budwood from a good tree. Since you mention having little space you could even make each branch a different type of citrus. The attached link gives pretty good examples of how to do it.

Here is a link that might be useful: grafting link

Here is another budding link. But please please please note the first sidebar. Taking budwood from just any healthy looking tree can be very problematic.

Here is a link that might be useful: budding link