Why can't I inject the ProductRepositoryInterface in Magento 2?

This generally crops up when you add another parameter to the constructor because Magento caches it in var/generation. You need to clear var/generation which forces Magento to regenerate the interceptor.

It's hard to say without further details, but I'm going to assume youe code sample actually looks something like this

namespace Packagename\Module\Controller;
class Test extends Action
{

    private $_productRepository;

    public function __construct(
        Context $context,
        StoreManagerInterface $storeManager,
        ProductRepositoryInterface $productRepository    
    ) {
        parent::__construct($context);
        $this->_storeManager = $storeManager;
        $this->_productRepository = $productRepository;
    }
}

i.e. with a PHP namespace (since all controllers have namespaces in Magento 2).

If that's the case, then in your code sample you're actually trying to inject a Packagename\Module\Controller\ProductRepositoryInterface class. If you use a class name without a namespace prefix, PHP assumes you want a class in the current namespace.

I was not able to inject any module to a controller with this error message. Deleting var/generation worked for me..