Is it possible to add bundled products with Magento API? If not, is there any other way to add a bundled product?
Yes there is way around, which I myself have been playing with for some time. Please see this post (Programmatically add Bundle Products in Magento, using the SKU / ID of Simple Items) to get a starting point of how to proceed.
I have used simple way to add Bundle Products, not by using Web Services. However, I think the latter is still possible.