Yes it is very possible for a person to pursue 3 Bachelor Degrees and 3 Masters in a lifetime. There can be many ways to pursue this, let me share with you one which strikes my mind.

Lets us assume Mr. X he does his B.Sc. in Chemistry from a reputed University like Calcutta University. After B.Sc. he becomes so much interested in Chemistry and its Applications instead of going into professional life he decides to pursue a M.Sc. from a Top Technical Institutes like IIT Kharagpur. Hence, he does his M.Sc.

After his 2 year M.Sc. he joins a Processing and Manufacturing Organization as a Research Associate. During work he realizes, his degree was more or less focussed on Core Chemistry but the work he does requires a lot of Material Science in it so he decides to pursue an Associate Degree in Metallurgical and Materials Engineering ( Equivalent to B.Tech ) from Institution of Engineers, hence he completes his B.Tech and joins Work at his Laboratory.

Three years later, he notices he wont get a Promotion in his Job from Associate Manager to Senior Manager unless he has a Postgraduation.

He appears for GATE Exam again and gets an Admission for M.Tech in Metallurgical Engineering from his alma mater IIT Kharagpur. He returns to campus on a sabattical from the Research Laboratory and completes his Postgraduation. Followed which he joins work and is also granted a Promotion.

As years pass he realizes the need for a MBA to secure Higher positions in the Organizations. He further completes his MBA from a reputed Institution like XLRI Jamshedpur follwing which he gets a Deputation as General Manager of Technology in the Organization.

Later he is promoted but Deputed as Senior General Manager of Corporate Affairs with the same company but he realises the need to acquire Legal knowledge to understand legal and corporate affairs, agreements of the company. He then decides to pursue a Bachelor of Law ( LL.B) Degree in Correspondence from IGNOU.

Hence, you see he has completed 3 Bachelors and 3 Masters Qualifications. They may be multiple other ways to do this.