Nisaa
No. Fasting is invalid for as long as hayd is present, whether the fast is an obligatory Ramadan fast, a voluntary one, or a day of qada owed from a previous Ramadan. This is one of the few rulings the four madhabs hold with no disagreement at all: menstrual bleeding stops a fast completely, and nothing she does can make that day's fast count while she is bleeding.
The Prophet (Salla Allahu 'Alayhi Wa Sallam) put the ruling to the Companions as something they already knew: "Is it not the case that when she menstruates she does not pray or fast?" They replied, "Yes." He said, "That is her shortcoming in religious commitment." (Narrated by Muslim). It is stated as a settled fact rather than a position he was arguing for, which is part of why later scholars treat it as agreed rather than disputed.
Fasts are made up, prayers are not
'Aishah (Radi'Allahu Anha) said: "When we would have our menses during the lifetime of the Prophet (Salla Allahu 'Alayhi Wa Sallam), we were ordered to make up the days of fasting that we had missed but were not ordered to make up the prayers that we had missed." (Narrated by al-Bukhari), the same hadith making up fasts missed for your period sets out in full. Fasting and prayer are not treated the same way: replacing every prayer missed across days of hayd would ask an unreasonable amount of a woman, while a missed fast day is a single countable unit, and Allah commands exactly that kind of make-up: "and whoever is ill or on a journey, the same number [of days which one did not observe Sawm (fasts) must be made up] from other days." [Al-Baqarah 2:185].
It makes no difference how far into the fasting day the bleeding begins. If hayd starts at any point before Maghrib, even minutes before it, that day's fast is invalidated from the moment bleeding starts and is owed afterward like any other missed day.
Why hayd stops a fast and Janabah does not
The fast is broken by menstrual bleeding itself, not by impurity in general, which is why different states are not all treated the same. Nifas, the bleeding that follows childbirth, stops a fast the same way hayd does, for as long as it continues. Janabah works the other way around: a state of Janabah still present at Fajr does not invalidate a fast at all, because it only requires purification before the next prayer, not before the fast can carry on. Istihadah, irregular bleeding that is not classed as menstruation in the first place, does not stop a fast either; see can you fast during istihadah for why the bleeding has to actually be hayd, not merely present, before it invalidates the fast.