There have been lots and lots of studies. The problem with the studies is that they all suffer from a phenomenon in social sciences called the Hawthorne effect. It's named after a town called Hawthorne, Ind., where a group of power-plant workers were being studied and they showed marked improvement just because they were being studied. The Hawthorne effect says that when a group of people is being studied, the most influential factor for any improvement in their activity is the attention they are given by the people who are doing the study. Just the fact that in all studies involving kids and computers you have the Hawthorne effect makes it difficult to separate the positive effects of giving kids computers from the positive effects of giving kids more attention.
Various studies show that kids improve their learning using other techniques, without computers.
Some schools have demonstrated high performance in learning with a de-emphasis on computers. We really don't know whether computers improve learning or not. I wouldn't say we'll never know, but the factors going into learning are so complex and interrelated that it's hard to determine. Teacher training is essential, but currently undervalued and under-budgeted. Another important feature that is undervalued is technical support for teachers and kids.