Yes, TurboTax Premier can account for multiple 1099-Bs from various firms. All gains and losses for passive activities are netted properly.   Wash sales are supported, unless they are related to employer stock sales. 

While the software handles all common transactions, there are a few  situations reported on form 1099-B that cannot be accounted for. These include:

• Nominee distributions 
• When Box 2b of Form 1099-B is checked, from a reportable change in control or capital structure reported in Box 2a of that form.
  
• The calculation of a disallowed wash sale loss adjustment on Form 8949 when employer stock is sold. Employer stock is stock acquired through various employer-sponsored plans, such as an Employee Stock Purchase Plan, an Incentive Stock Option Plan, a Restricted Stock Plan, or a Restricted Stock Unit Plan
  

The following contracts can be entered manually, but are not supported for import: Regulated Futures Contracts, Foreign Currency Contracts and Section 1256 Option Contracts.

Hello AmandaR1, 

Do I understand you right: Turbotax Premier can handle wash sale from different brokers even if option trading is involved(but not employee stock)?

No 1040 tax software will correlate trades of the same security at different brokers that result in wash sales.

That will be your responsibility.

In general, for your own sanity, don't do that.