Footing the bill likely turns on whether the LL can establish that the tenant introduced the BBs to the property.
As always, the first place to reference a lease--is the lease agreement to see if it has a provision that applies that does not offend state law.
In most states, the LL is responsible to exterminate pests, the tenant is responsible to report them--in writing. If the LL can then show that the tenant is the one who bought the pest into the property, the LL can then attempt to recover costs.
I recall that Indiana ( I am NOT an Indiana attorney) is one of those states that does not clearly place responsibility on one party or the other. A tenant who can argue that the LL has failed to maintain a warranty of habitability, or violated a local housing or safety code MIGHT stand a shot at recovering costs in Indiana.
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Unlikely. The landlord has the duty to provide a habitable unit but not to maintain it that way during your time there. That duty is typically the tenants. The "but" here is that if this is a multi unit facility and something the landlord is doing (or not doing) as a whole is facilitating the infestation. Look for the "maintenance" clause in your lease. It likely says you will maintain the premises.