No, balance transfers do not increase your credit limit. You cannot transfer a balance that exceeds your account's credit limit, and issuers will either reject such a balance transfer request or accept only a partial transfer.
Many credit cards offer introductory 0% APR periods on balance transfers to new cardholders, and you can often request the transfer on the application itself. Unfortunately, if you request a balance transfer as you're applying for a new card, you won't know your exact credit limit until after your application is approved. However, some issuers may periodically extend balance transfer offers to existing cardholders. These offers usually disclose your balance transfer credit limit.
It's important to note that most balance transfer offers include a balance transfer fee. The fees are usually 3% to 5% of the amount transferred, so make sure you have enough available credit to cover the cost of both the balance transfer and the transfer fee.