Can submitting multiple health insurance applications hurt me?

[Mommy_Again]

I am currently COBRAing through my old job at a cost of $965/month and need to find private insurance. Because of some medical issues, I suspect the rates I am being quoted will increase once the application is completed, so I need to get firm rates. Is there a downside to submitting applications through multiple insurers? The only thing I can compare this to is having credit reports pulled often and getting dings on your credit score. Can this happen with health insurance quotes?

TIA!