Technically you can start filing from today, but practically it is not possible.
If you are salaried, login to ITR website and see if last quarters (Jan Feb Mar) tax is paid by your employer or not.
You will see that it is NOT. Because employers will file it only by end of April-May and it will reflect in your AIS by June only.
Same is the case for interest earned in last Q, banks will report it but it takes time to reflect in AIS.
All in all, it takes 2 more months for all your transactions to flow in to AIS. By end of June, you should see all your income in AIS - salary, interest, MF redemptions, share sales, dividends, land sale, etc.
So better wait till July first week and then file.