Catalin, it depends on various factors which include amongst other things: the standard practice, the enforcement agency, job specifications and purpose of the test.
I am familiar with ASTM C 31- Standard Practice for Making and Curing Concrete Test Specimens in the Field. This standard is referenced in ACI 318-Building Code Requirements for Structural Concrete and Commentary. For most cast-in-place concrete building structures, ASTM C31 is the standard practice adhered to in the United States
According to Section 3.2 of ASTM C 31, final curing of strength test specimens(after initial field curing) in a lab curing room or tank filled with lime saturated water is done for reasons such as:
1. Acceptance testing for specified strength
2. Checking adequacy of mix proprotions for strength, and
3. Quality Control
Sec 3.3 deals with the situation you describe..i.e field curing adjacent to structure for reasons such as:
(i) Determination of the time the structure is permitted to be put in service
(ii) Comparison with test results of standard cured specimens or with test results from various in-place test methods,
(iii) Adequacy of curing and protection of concrete in the structure, or
(iv) Form or shoring removal time requirements
For Highway projects, state departments of transportation use AASHTO standards; so instead of ASTM C 31, they use AASHTO T23--Making and Curing Concrete Test Specimens in the Field..and typically make ammendments to this document. So while some DOTs require that specimens be cured in lab after initial curing in field, others may not. Example:
see Section 7
refer to Section 9