I will turn down my beliefs and say that humans do have a soul in this scenario, therefore I can answer your question properly.
I would strongly say no, just because a robot becomes sentient and cognitive does not give it a soul. I think we could agree that in this scenario, if souls are real then we could assume that there is some form of god or deity without sounding to presumptuous. In that case I would think that this god or deity would limit souls to only natural beings e.g. Humans, tigers, lions, bears etc.
It doesn't exactly seem prudent to give robots, even sentient and cognitive ones, souls. I know that may sound too harsh, but if you think to the near future, robots like that will start being made every day in factories so they can do all the tasks we don't want to do. Imagine if every computer on Earth was given a soul, there would be far too many of them to handle for that god or deity thing.
Now thinking harder, it seems to answer this question we must define soul, cutesy of Google definitions I found this.
Define Soul:
the immaterial part of a person; the actuating cause of an individual life
person: a human being; "there was too much for one person to do"
deep feeling or emotion
the human embodiment of something; "the soul of honor"
Notice the third definition please, so according to these definitions it may well be possible for advanced robots to posses souls. I would still say no, even if souls are real, but I would definitely have to not disagree with them having souls quite as strongly as before.