I just got a call from the same number, but I didn't pick up.  No voice mail left either.  Didn't try calling back.

I just received it as well but didn't pick up.

I got two calls from (866) 597-4938 in the last 24 hrs.

I called it back and it goes straight to a vmail.  "Thank you for your interest in verizon wireless.  If you are calling with questions regarding a recent bill you received or have technical questions, please hang up and dial 18007357744.  For sales inquiries only, press 1 and a sales rep will be with you shortly."

??  sounds like spam to me....

Got the same thing called back was on hold for 30 mins gave up called verizon directly had no idea of any upgrade available on my acct

UPDATE: I posted the above message about the FCC. In response to the complaint that I filed through the FCC, I received a confirmation letter from the FCC, and shortly after, a call from Verizon on my home phone inquiring about the phone calls I have received on my cell phone. I was a bit suspicious hearing from Verizon, considering I've been receiving these other strange calls.

The caller indicated that Verizon received the complaint from the FCC and a person there is investigating the phone number. I was asked some quick questions and was told that I will be contacted again soon with results of the investigation.

They called me and said they were verizon.  The girl sounded like an idiot and was reading from a script.  I told them to not call me during normal business hours while I'm at work...yet they just called me again at 11am and did not leave a voicemail.

Got a call today from this number.  Supposedly a Verizon Wireless representative.  They had info about my cell phone account, and it sounded legitimate.  It very well might have been.  She wanted me to upgrade my phone, and it sounded convincing.  I didn't do it, but I got the phone number, called it back, and I keep getting a weird message that doesn't sound right saying "please call back during normal business hours to speak with a representative."  
Anyone else get a call from this number?

I have received many calls from this number on my cell phone - I don't usually pick up. In frustration, this morning I did answer the phone (8:15am on a weekday), hoping to tell them to stop calling me. The call ended as soon as I answered the phone.

I called the phone number back and got voice mail. The voice mail message indicated that the company is Verizon Wireless, gave a different phone number for questions about billing or technical issues, and then said to press "1" for sales inquiries. I pressed "1" and waited several minutes, then heard the following, "if you would like to make a call, please hang up and dial again."

I filed a complaint at the FCC website.
http://esupport.fcc.gov/complaints.htm

1) Selected the option on telemarketing
2) Selected the option appropriate to you (I selected the option that included "Abandoned calls on wireless phone")
3) These options brought me to the "FCC Form 1088G - Call or Message to Wireless Device (Cell Phone or Pager) Complaint."
4) I filled out the info and sent it electronically