A Thought Experiment by and on You
Please conduct the following simple (series) of thought experiment(s) by and on yourself (X): Take your actual (current) date of birth (DOB(X)=x) and now construct/compute a date exactly one year earlier than your DOB. Got it?
Ok, now throw away your old DOB and focus on this new, derivative date (x-1) and try to imagine that x-1 is your DOB. Got it?  .
The question to ask yourself now: Would this person, X, born on this new DOB, x-1, but ‘ceteris paribus’, be still you, X?
Once you have an answer (yes, no, don’t know, it depends …) take now the date exactly one year later than your original DOB, x+1 and ask the same question.
Now make the difference +/- 2 years, 3 years, 4 years, 5 years … and ask the same question. Stop when you get the same answer 2-3 times in a row or notice if you change your position with different numbers. Register whether you think differently if the test scenario is earlier or later than your actual DOB.
In what follows I will delineate some potential answers to this thought experiment. These answers won’t follow any derivative historical positions and only grew out of me thinking about the possible logical space and interviewing quickly several acquaintances of mine, yet they (probably) might coincide with some actual philosophical accounts that could be given. I am looking for ways to make the scope of these possible answers comprehensive, to hash out the logical space.
I start with the conservative, negative positions to move towards the more permissive ones.
The negative answers
I. The anti-modalist, categorical naysayer
The first, unimaginative answer is literally the ‘unimaginative’ one. One that does not allow running possible world composing, counterfactual thought experiments to test heavy philosophical concepts like personal identity. This anti-modal interpretation could be thought of as the fatalistic one perhaps, one that assigns some necessity to the already existing state of affairs. No point arguing against this one here just yet, would block further exposition.
II/A The neutral conventionalist to whom DOB does not matter
This answer revolves around the idea that since numbering the years is a human convention, being off years plus/minus compared to a date (we arbitrarily call our date of birth) based on a convention does not make a difference at all concerning the nature of this contract we call our existence. Please note that this position is, by default quite neutral in terms of other underlying issues, stops at a surface point.
II/B. The sceptical conventionalist and hardcore time-world relativist
But maybe the conventionalist position can also be turned into a full blown sceptical account concerning the meaningfulness and futility of such a thought experiment. If the DOB is based on a convention, then modifying this parameter is not going to make a difference in the fabric of the supposedly affected possible worlds conveniently close to our actual world? The point being is that there’s no fix point that can be picked out at the intersection of times and worlds that do slide over each other, so all these positions are so relative to each other that one year later or one year earlier will leave the worlds intact, but not distinct from each other. If we rewind the clock of the DOB in possible w’ compared to actual word w, everything else still stays the same. This position merits further attention but goes deeply into how we think of possible worlds and how we think on the parallels of worlds and times. But we leave satisfying this attention to another occasion.
III. The biological no answer
Those who wish to maintain a strong ‘biological identity’ can argue that the same combination of oocyte and sperm that formed the subject/object person X of the thought experiment cannot possibly meet and form the zygote one year earlier or later. As far as oocytes concerned, they are ‘for life’ meaning that normally women are born with a stock of oocytes in their ovaries, couple millions, and this reservoir gets depleted throughout their lives with usually no viable oocytes remaining at menopause. So your particular founder oocyte could back you up for decades.
But sperms are not for life and they don’t come as a starter pack at birth of a man. Most importantly they usually last only couple tens of days, one such number provided in the literature (floating around in the internet, but will check it stricter later, after all am a biologist by default) is 74 days.
And that is bad news concerning the idea of being born a year later or earlier, since a year is at least 365 days. On the other hand since 365 is only ~5 times as big as 74 days, for those biologically inclined we can make an exception to keep them in the philosophy game and limit this thought experiment to days or tens of days instead of years and ask whether X thinks X could have been born DOB+/-10, 20, 30, 40, 50 days later/earlier. This way, the ‘biological no’ answer objection is not valid and can be compatible with a more permissive position.
Another logical possibility is to assume not just that I have been born years later or earlier, but for the actual sperm participating in my birth…
Although bodily identity and biological identity is a broader concept than zygotic identity, which is really the point here, but we assume here (probably safely) that zygotic identity is a necessary condition of both biological and bodily identities.
IV. The predestinationist
From biologist (possibly physicalist, but that might warrant a separate look) waters we are arriving to more spiritual/religious waters, when the concept/doctrine of predestination fixes one random sequence in space-time for a life history of a particular person and part of this package is the DOB of X, fixed once and for all. From this point of view the question of modulating this DOB might be a no-go, even without considering how this might affect personal identity, or even having a detailed account of personal identity within this framework. One problem a predestinationist encounters is how to reconcile their central doctrine with the outright convention underlying the Gregorian calendar, or any other calendars as a matter of fact.
Yes and no: temporal asymmetry as the answer in between
This position can be extracted from Identifiable individuals by Arthur N. Prior, accessible in Papers on Time and Tense, originally published in 1960 in the Review of Metaphysics. It says, that while it is impossible for a particular X person (past or present part of the actual world) to have been born before the DOB registered in the course of the actual world, it is still possible that he could have been born couple years later. Concerning the first impossibility part the argument goes like this: the time period before X has been born, makes X indistinguishable from any other people not been born before the actual DOB. of X. Hence considering X’s pre-existence before X’s actual DOB is not going to be able to pick out and individuate X from other non-borns. But once we have a particular X born at a particular DOB it can be considered without internal inconsistency that X could have been born couple years after their actual DOB, since there’s a particular continuant to make counterfactual statements about.
The positive answers will be discussed in part 2 of this study. A positive answer is the position where conducting my thought experiment landed me originally, also this is the position that opened up my imagination.
 I realise that the way the question is framed is already indicative of an affirmative position concerning its feasibility, but see no other way to start seriously thinking about it as suggesting impossibility so early makes one’s mind mute.