As long as you have a good credit score and meet the eligibility criteria for either of the loans, you can avail them. The lender has to be convinced enough to know that you would be able to handle both the loans and make payments regularly. Juggling between both the loans might be a tough feat, but as long as your income is able to cover your monthly payments, you will be able to do it.
Before applying for either of them, you can have your priorities sorted on, as to which loan you would need at the latest. Find out if you will be able to pay both EMIs every month without difficulty. The lenders will check into your credit history to make sure that you haven’t defaulted in any of your other credit payments. If you have any debts lying around it is better to clear them off as, if your going for two loans you will definitely end up getting both the loans rejected. Personal loans will have higher interest rates than home loans, as personal loans are unsecured loans. If your credit score is good, you can avail a personal loan with less interest rates.
So applying for both a home loan and a personal loan can be done as long as you are eligible and if the bank deems you fit for handling two loans at the same time.
Additional Reading: Can I get a personal loan if I have home loan?