If you had a sequence 625 or more 32-bit integers, you could detect with high confidence whether it was from consecutive calls to Mersenne Twister. That is because it leaks state information in the output values.
For an example of how it is done, see this blog entry.
Similar results are in theory possible when you don't have ideal data such as full 32-bit integers, but you would need a longer sequence and the maths gets harder. You would also need to know - or perhaps guess by trying obvious options - how the numbers were being reduced from the larger range to the smaller one.
Similar results are possible from other PRNGs, but generally only the non-cryptographic ones.
In principle you could identify specific PRNG sequences with very high confidence, but even simple barriers such as missing numbers from the strict sequence can make it a lot harder. There will also be many PRNGs that you will not be able to reliably detect, and typically you will either have close to 100% confidence of a match (to a hackable PRNG) or 0% confidence of any match.
Whether or not a PRNG is a hackable (and therefore could be detected by the numbers it emits) is not a general indicator of PRNG quality. Obviously, "hackable" is opposite to a requirement for "secure", so don't consider Mersenne Twister for creating unguessable codes. However, do consider it as a source of randomness for e.g. neural networks, genetic algorithms, monte-carlo simulations and other places where you need a lot of statistically random-looking data.