This usage presumes the existence of a schedule in which days are titled by number, perhaps escalating, perhaps not. The days might or might not be in a countdown to somebody's birthday, for example. This could cause confusion. The 5th week of pregnancy indicates that the foetus has been growing for 5 weeks, not that there are now only 5 weeks till the child is born.
Sometimes certain days might not be included. For example, in a TV sitcom shown everyday except on weekends. It begins on Monday and runs till Friday every week.
Day Six of the season is a Monday, not a Saturday, since the series is not shown on weekends.
A distinction might need to be made between Day 6 (Monday in the 2nd Week) and the sixth day (Saturday of the 1st Week, a day on which an episode is not shown). This could be necessary for those involved in producing the series for television. The production team might need to keep in mind two calendars that run concurrently: one for the characters in the show (Day 1, Day 2...) and another for the production team who need to know when their own holidays are. (You have a day off on the sixth, seventh, thirteenth and fourteenth days.) Maintaining a system in which the different calendars are referred to in distinct ways can avoid confusion.
There is an ISO standard, ISO 8601, which is used in business, for example, to specify week numbers. You could indicate Week 50 as follows:
Or including a day too:
One advantage of this usage is that it is possible to tell directly from the date the day of the week, too. In the above example, it is the fifth day of the week, Friday, as indicated by the 5.
(There are disadvantages, too. For example, 2014-12-31, the 31st of December, 2014, is indicated using ISO 8601 as 2015-W01-03. Though this day falls in 2014, it is shown as part of Week 01 in 2015.)
The Fourth Day
Stylistically, one might not wish to lose the poetry of language by inclusion of digits. Compare the following:
On the third day, he rose from the dead.
On the 3rd day, he rose from the dead.
The Fourth of July
Day 4 of July
When one is presenting information in such a way that it would be useful for numerical data to be gleaned at a glance, using numerals instead of letters is helpful, as it highlights, through contrast against a background of letters, the most salient information and lends itself more readily to inclusion in a table.
Day 1: Boy Meets Girl
Day 4: They Fall in Love
Day 7: They Get Married
Compare this with the less aligned:
The First Day: Boy Meets Girl
The Fourth Day: They Fall in Love
The Seventh Day: They Get Married