Can you? maybe.
Is it legal? no.
Are you likely to get caught? Moreso now than in previous years, the interstate compacts are getting pretty fast at catching things like this.
Not legally. Good luck.
DISCLAIMER I do not practice law in your State. This answer is provided for informational purposes only. This answer does not constitute legal advice, create an attorney-client relationship, or constitute attorney advertising.
With respect to the other folks who have answered your question, it depends on what kind of suspension we are talking about. The Interstate Driver's License Compact (agreement between states that governs the recognition of each other's DL suspensions) only says that the member states shall recognize suspensions that come out of convictions for offenses. It is silent as to suspensions that come out of administrative license revocation hearings and suspensions for failure to pay surcharges. That means it is up to the individual state to decide whether or not they want to recognize the suspension. You can always go ahead and apply for the non-Texas license so long as you answer all questions truthfully. They will either approve or deny your application.
Keep in mind, that even if the state you get the DL out of gives you a license, your privilege to drive within Texas will still exist (unless of course you resolve that issue) because you are in Texas. You may be eligible to get an occupational driver's license on a Texas license. Talk to a local criminal defense attorney who handles ODLs for more information.
Disclaimer: This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.