After a serendipitous discovery of bright [Fe ii]lambda16435 emission in nebulae around eta Carinae and P Cygni, infrared spectra of other luminous blue variables (LBV) and LBV candidates were obtained. Bright infrared [Fe ii] emission appears to be a common property among LBVs with prominent nebulae; this is an interesting discovery because strong [Fe ii]lambda16435 is typically seen in shock-excited objects like supernova remnants and outflows from newly formed massive stars, as well as in active galactic nuclei (AGN), where the excitation mechanism is uncertain. This paper presents spectra in the H-band (1.5 to 1.75 mum) for the central stars and nebulae of eta Car, AG Car, P Cyg, Wra 751, HR Car, HD 168625, HD 160529, R 127 and S Doradus. Seven of nine targets show bright [Feii]lambda16435 in their nebulae, while it is absent in all central stars except the LBV candidate Wra 751. The two objects (S Dor and HD 160529) without prominent [Feii]lambda16435 are not yet known to have nebulae detected in optical images, and both lack bright thermal infrared emission from dust. The possible excitation mechanisms for this line and the implications of its discovery in LBV nebulae are discussed; there are good reasons to expect shock excitation in some objects, but other mechanisms cannot be ruled out.