Anxiety can cause symptoms that look like serious health problems, for sure. I've had panic attacks before, and even though I was familiar with the symptoms of panic attacks, I still didn't recognize them at first. I thought I was having a medical problem, and that I might have to go to the hospital. What's weird was, I didn't even think I was anxious, at the time.
I've also had a panic attack that was triggered by paying too much attention to my heartbeat and worrying that I might have a heart problem. That time, it actually felt like my heart suddenly stopped beating for a few seconds, and I felt this wave of weakness come over me, and I became so weak that I literally could not move or even lift an arm for several seconds, and I felt like I might go unconscious. But even that turned out to be just anxiety and completely harmless, and it hasn't happened since.
I've heard something that can help for people who have anxiety like this is to try inducing symptoms, like doing shallow breathing, so you can experience the physical sensations that make you anxious and realize they're not harmful. It's like exposure for a phobia.