I know this thread is somewhat outdated, now, but I'm new to Engineering Tips and am searching some of my favorite topics. I am an independent consultant that makes my living using various FEA packages and let me say that just because one of us hasn't done something, it doesn't mean it can't be done. You can mesh individual parts in Algor and the mesh will line up, or you can mesh the entire model and the mesh will line up...if you do it right. With that said...
Brad made some vary good points and, fortunately, recognized his own limitations in certain areas. I have used Algor, NEiNastran, COSMOS, and many specialty packages. Algor continues to enhance their user interface, but their engine is sound (including non-linear). "Bang for the buck" Algor is hard to beat. NEiNastran is good competition as is COSMOS. Each have their strengths.
I am generally leary of anyone that says "Stay clear of ..." This is a competitive industry. If something didn't work, and wasn't getting supported technically, the package would have long since been out of business. NEiNastran is based on the Cosmic Nastran kernel. A good package, but it is still maturing. Teaming with FEMAP was smart because that gives it CAD capabilities and the ability to translate to a variety of packages. COSMOS has a very sound engine. Now that it is owned by the same company that owns SolidWorks, we will likely see continued growth in the interface beyond the current COSMOSWorks that is available. Algor has remained fairly independent. The greatest advantage, perhaps, is that there is one person to contact for all your needs, and they are very responsive. As for the question about highly non-linear materials, all of these packages (which I would consider mid-grade) have Odgen strain models in addition to the general hyperelastic models. The analysis code is well-understood and well implemented.
For a model your size, the p-element convergence codes (like Mechanica) are generally very effective. They reduce the number of nodes (and, therefore, calculation overhead) by orders of magnitude.
The higher end codes, like Mechanica, Ansys, and Abaqus, are excellent if you need to afford them. They are great for fracture mechanics and specialty applications, which it doesn't sound like yours qualifies. As for limitations on any of these or the other packages, most of the limitation is in your hardware, not the software.
I'm all for gaining additional capability, but remember, I'm a consultant that has to interface with many different people using a variety of packages...if you have the tools, learn how to use them, don't just assume you need new ones.