Bronze, silver, gold, platinum refer to the actuarial value of a plan. Generally speaking, actuarial value is the overall percentage an insurer expects to pay for all treatments under that plan for a "normal" consumer, not counting premiums. E.g. a silver plan (70% actuarial value) expects to pay 70% of all treatment cost for a normal consumer in a plan year, including deductibles, out of pocket maximums, coinsurance and copays but not including premiums.
A doctor that is part of a network plan agrees to a certain contracted price for a service, ceterus paribus, regardless of the actuarial value of the insurance used. The split of the service cost between insurer and consumer is what changes, not the provider's contract price.
Limited network plans ("skinny networks") are a phenomenon of PPOs and PPO hybrids (since HMOs are generally exclusive provider networks). PPOs traditionally do not use a gatekeeper model, meaning that you do not need a referral to see a specialist. Under the PPO model you get the in-network price so long as you see any qualified provider in the network, but that does not mean that all providers are under the same contracts. A highly desirable medical group may be able to command a larger contract price with an insurer, but the insurer is going to pay the same percentage (since most PPOs use coinsurance).
Example: A plan has a 80% coinsurance after deductible. There are two specialists in the network, one a highly desirable medical group and the other a new practice with few clients. The insurer contracts with the new practice to provide a specific service for $100; the insurer will pay $80 for that service if the insured's deductible is met and $0 if it is not. The medical group has leverage and gets the insurer to agree to a $200 contract price for the same service; the insurer will pay $160 for that service if the insured's deductible is met and $0 if it is not. The insurer will pay 100% of the cost in both cases if the insured has met the annual out of pocket max.
***WARNING: MATH FOR ILLUSTRATIVE PURPOSES. NOT ACTUARIALLY SOUND***
The insurer anticipates 2000 instances of the service being rendered in the plan year.
500 at the new practice (@ $100 ea)
1500 at the medical group (@ $200 ea)
At the new practice:
50 occur before the deductible is met (total insurer cost: $0)
375 occur after the deductible is met (total insurer cost: $30,000)
75 occur after the OOP max is met (total insurer cost: $7,500)
Total cost at the new practice: $37,500
At the medical group:
425 occur before the deductible is met (total insurer cost: $0)
980 occur after the deductible is met (total insurer cost: $156,800)
95 occur after the OOP max is met (total insurer cost: $19,000)
Total cost at the medical group: $175,800
Total cost to the insurer: $213,300
Total insureds: 10,000
Premium cost per insured per month: $1.78
***END OF ILLUSTRATIVE MATH***
Now, take that same scenario but remove the medical group from the picture (i.e. a skinny network). All of the medical group loss costs are gone and the insurer loses the 8,000 enrollees, leaving 2,000 people to accumulate $37,500 in service expenses. That a pmpm (per member per month) rate of $1.56. By skinnying up the network the insurer reduces premium by $0.22 pmpm. That's good for hitting the 60% and 70% actuarial values of bronze and silver plans because the rules allow deductibles and whatnot to be manipulated only so much. Since gold and platinum require higher actuarial value the expensive providers can be added back in to boost the value.
***WARNING: MORE ILLUSTRATIVE MATH***
In the example above, with 10,000 members, the insurer had 500 instances of the service being rendered at the practice and 1500 instance at the medical group, for a total of 2,000 instances. The total cost of all instances was $350,000 (500*$100+1500*$200). The insurer paid $213,300. The insurer paid 60.9% of the cost, which is bronze level.
Removing the medical group, there were 500 instances of the service being rendered at $100 each for a total cost of $50,000. The insurer paid $37,500. The insurer paid 75% of the cost, which is between silver and gold.
***END OF ILLUSTRATIVE MATH***
Wait a minute, you say. Doesn't that go against the example? Not really. Look at the incidence of occurrence. The practice had 500 services and 2000 enrollees, so 1 in 4 enrollees received the service. The medical group had 1500 services and 8000 enrollees, so 1 in 5.3 enrollees received the service. Additionally, the practice enrollees were more likely to reach the OOP max than the medical group enrollees (15% v 6.3%).
Moral of the long-winded answer: premium rate setting is fucking complicated and attempts to distill it down to a sound byte play well in the media but are really irresponsible.