NP's in general are able to perform procedures that are considered invasive with proper training. Federal regulations allow NP's to perform procedures that a physician does and actually provide for a mechanism wherein the NP can be reimbursed for these services. However, the extent of how invasive a procedure an NP is allowed to perform depends largely on individual BON regulations as well as individual organizational protocols within hospitals and group medical practices.
Acute Care NP's are best poised in terms of receiving training for the kind of procedures you mentioned because it is the high acuity patients in hospitalized settings who typically require central lines, chest tubes, and LP's. As an ACNP who works in an ICU setting, I have actually placed all kinds of central lines (from PICC's, non-tunneled dialysis catheters, multi lumen CVC's, and Swan-Ganz) and a few chest tube types (thoracostomy tubes, CASP). There are a few changes going on in healthcare that is driving the trend that NP's are now doing invasive procedures such as these.
In academic and other types of teaching hospital settings, one such change is the limitation imposed on the amount of continuous hours residents or house officers can remain in the hospital. Because of these limitations and the fact that residency slots have not increased, NP's (or PA's) have been called upon to supplement the manpower required in maintaining 24-hour coverage of ICU's, ER's and inpatient units in hospitals. Thus, NP's have also been trained to perform invasive procedures typically performed by residents because that is part of providing care to the patients in these settings. Note that attendings in these settings do not typically do these procedures themselves.
The other factor that have given rise to this trend is the shortage of intensivists and other types of specialists particularly in remote regions and rural areas of the US. In these settings, NP's have assumed these skillsets in order to supplement the smaller number of available practicing physicians in meeting the healthcare needs of high acuity patients.