My US visa application has been refused, but I still have an old valid visa. Can I proceed to travel to the US?

The CBP will see any visa refusals in their system, so even if the visa officer didn't cancel your old visa (which he/she should have done), the simple fact is you will not get in.

Firstly because, according to protocol, the old visa should have been cancelled when you applied for a new one - in other words, you should not have any valid visa at this point.

Secondly for the obvious reason that the CBP uses almost the exact same criteria to determine admissibility as the visa officers do to determine visa eligibility.

So no, just no, do not try this, because you'll be sent back and probably barred from re-entering for a certain period of time. If you're lucky

Don't risk it. In October 1999 I had essentially the same situation. My visa was a couple of weeks from expiring and I sent it in for a renewal which was supposed to be without an interview. I was promptly called for an interview and the renewal was denied and my passport stamped to indicate a refusal. I believe it was because after college I came over for a 5 months vacation in the USA instead of the two months I had indicated I would spend.

I was working on catching a plane to the USA within the four days validity remaining on the visa since it had not been canceled (or at least not visibly cancelled) but couldn't make it for one reason or the other but mainly because I had a pretty much close to guaranteed opportunity to come to the USA the following year for grad school.

A year later at the interview for my students visa I asked the consular if I could have entered the previous year as I planned to without any problems. He smiled and informed me if the airline allowed me to board (I think it was before the days when airlines could check the validity of visas as they can now) I would have been refused entry because the visa had been voided in their Consular Lookout and Support System (CLASS) and I would have jeopardize my opportunity for the students visa.

Usually when you go to apply for a new visa, they cancel the existing visa (e.g. write "cancelled without prejudice" on it). If they didn't do that (which is weird), then technically, yes, you can use it to try to enter the US, on or before the visa's expiration date.

However, just because you have a valid visa doesn't mean the immigration officers at the border will let you in. The visa just entitles you to travel to the US to apply for admission. The immigration officer makes their own determination of whether you are admissible every time you seek to enter. You did not say what reason was given for your visa denial, but whatever it was, if it still applies, the same reason would be a reason to deny you entry. The immigration officers will likely see the visa denial in their system and be alerted to look out for whatever issue caused your visa denial.