Theoretically I reckon it could be possible but it would cause a number of problems that we would have to get around..
The white blood cells in your body are strongly regulated by other cells (and by each other) so that they do not overactivate or cause damage by reacting to self tissue. If you had a robotic cell that started acting a bit over the top – or got a mutation and became cancerous – you would have a bit of a problem because you wouldn’t be able to kill it or get it back out of the body. It would be like an immortal superhero that turns evil and spends his life getting the world in jeopardy. Also you would probably need more than one robotic cell to help fight an infection (because it would likely be specific for only one sort of disease causing pathogen and also it takes a while to get around the body) – but then you run into problems because if you give it the ability to proliferate (make more of itself) you could get overrun with the robots and if some of them turned evil you would have a disaster!!!
But its an interesting thought and certain derivations of this idea are certainly being thought about!
Theres a game you can play where you travel through the vessels and try to kill bugs! You have to navigate a nanobot through a 3D environment of blood vessels and connective tissue in an attempt to save an ailing patient by retraining her non-functional immune cells. Along the way, you will learn about the biological processes that enable macrophages and neutrophils – white blood cells – to detect and fight infections.