Hello and welcome.
I am not a doctor, but do have over 35 years experience managing chronic back pain. I would have to answer 'yes'. Pain from a spinal problem could radiate to the chest.
Please read through the Tip at the top of this WebMD Back Pain Community that lists the recommended steps for diagnosing and treating back pain. Within that tip are links to websites that have very good related information.
Since you notice the pain first thing in the morning, it may be your mattress if is is older. Many people find relief from back pain when they get a mattress that offers proper support. It could be something more complex, so seeing your doctor would be a good idea.
I hope you can answers, treatment and relief soon.
Click on my user name or avatar picture to read my story.
Blessings,
-Dave