First let's distinguish between the two different questions: The first question, to which you referred and which I answered, is "Can you receive Communion if you are late for Mass?" The second question, your question, is "At what point in the Mass does one's later arrival mean that one has not fulfilled a Mass obligation on a Sunday or holy day?" This thread will deal with the second question.
Prior to Vatican II, the common catechesis was that a person had to be present for the Offertory, Consecration, and Communion, or one had not fulfilled the Mass obligation and was required to go to another Mass. While a noble attempt to get people to church on time by giving them the time at which they were late, it had two unforeseen effects:
- Those with freer consciences would arrive after the start of the Mass, knowing that as long as they got there on time for the Offertory all was well. Although I am a post-Vatican-II convert myself, I have been told by pre-Vatican-II Catholics that it was not unusual for people to walk in after the Mass began but before the Offertory started.
- Those with tender consciences suffered deeply from scruples and would believe themselves in a state of mortal sin even though their tardiness to Mass was entirely out of their control (e.g., the car broke down; road hazards slowed down traffic; etc.).
Both of these conditions were unhealthy, and following Vatican II the cut-off point of the Offertory was dropped. Another reason that contributed to that was the elevation of the liturgy of the word and the homily to their modern importance in the Mass.
Basically, the Church wants us to be there for the full Mass:
§1 The obligation of assisting at Mass is satisfied wherever Mass is celebrated in a Catholic rite either on a holy day itself or on the evening of the previous day.
§2 If it is impossible to assist at a eucharistic celebration, either because no sacred minister is available or for some other grave reason, the faithful are strongly recommended to take part in a liturgy of the word, if there be such in the parish church or some other sacred place, which is celebrated in accordance with the provisions laid down by the diocesan bishop; or to spend an appropriate time in prayer, whether personally or as a family or, as occasion presents, in a group of families (canon 1248).
If there is just cause for being late to a particular Mass, one has still met one's Sunday obligation (and can receive Communion), but being late should not become a habit. If there is not just cause, one may still have met the Sunday obligation but the fact that one has not treated the Mass as a serious and holy event to which one should be prompt might be a matter to consider during an examination of conscience. If the matter is not mortally sinful because of lack of full knowledge or lack of free consent, one can still receive Communion. Because there is no longer a cut-off point after which you are late for Mass, the temptation to regularly budget one's time around the Offertory and skid into the pew just as it begins is removed.
When considering the question, our primary concern should be to be present for the re-presentation of Christ's sacrifice on Calvary. Those who were present that day two thousand years ago gathered around their Lord as quickly as they could and stayed with him throughout his agony, praying and suffering as well. Two thousand years later, we owe our Lord no less devotion than our forefathers and foremothers in the faith gave him then.