If I file with government & private companies can I collect from both?

Collect what for ? If it is for disability , or unemployment or other entitlements benefits , is illegal most of the time , collecting same benefits from two sources. This time you speak to your lawyer not your doctor .
Probably not. If you are talking about medical insurance, the best you can hope for is that the bill will be covered between the two policies. Usually one policy (often the government one) is considered primary and it pays the the majority of the bill. If there is any co-pay or balance, the other policy may pay all or part of that. If you try and collect the full amount from both insurers you risk fraud charges.