Can you file a lawsuit against someone if they have filed a lawsuit against you?

In general, yes.

Per the rules of civil procedure of many/most states, if your claim arises from the same underlying transaction or occurrence as the suit that was filed against you, it is considered to be a "compulsory counclaim". If not asserted within the context of the main lawsuit, it may be deemed to be waived and may not be brought as a separate suit.

In contrast, if your claim does not arise from the same transaction or occurrence as the main claim, it is likely to be deemed a "permissive counterclaim". As long as the individual who sued you is a proper defendant in that suit, the claim may, but does not have to be asserted as a counterclaim within the context of the main claim. That is, it may be maintained as a separate suit as long as the suit is filed within the applicable statute of limitations.

If there are third parties who need to be brought in as defendants so that there may be a full adjudication by the court of all the rights and obligations arising from the transactions or events, they should be named as defendants in the counterclaim, appropriate allegations made against them, and they must be served with process so that the court acquires jurisdiction over them.

The rules of civil procedure of the applicable jurisdiction will dictate the manner and mechanics of compliance with this general rule.