First, there’s no such thing as a “red flag” – just a low credit rating. Even then, some credit providers will give more weight to your credit rating than others.
To answer your specific question, it totally depends on your individual financial circumstances. Factors affecting how many cards you should apply for include the following:
Your individual and/or household income.
Your debts and your ability to pay them.
Your assets and liabilities.
The total credit limits of all your cards.
Your past spending and, in particular, your repayment history.
Your employment situation.
If you have a high income, low debts, low credit limits and a good repayment history, you can apply for more cards. Keep in mind, however, that you need to spend a certain amount of money with each card before you can receive the sign-up bonuses. I never apply for a new card before I’ve met the minimum spending criteria for the last card I applied for. And as a rule I try to cancel as many cards as I apply for. Cancelling a card will have a positive effect on your credit rating.
And of course you should never spend more on your cards than you can pay off in full each month!